<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Three-Dimensional Two-Pion Emission Source at SPS:
Extraction of Source Breakup Time and Emission Duration

P. Chung  Dept of Chemistry, SUNY Stony Brook, Stony Brook, NY 11794, USA  P.
Danielewicz  National Superconducting Cyclotron Laboratory, MSU, East Lansing,
MI, USA,
Dept of Physics & Astronomy, Michigan State University, East Lansing, MI, USA.
the NA49 Collaboration

###### Abstract

A model-independent, three-dimensional source function for pion pairs has been
extracted from Pb+Pb collisions at $\sqrt{s}_{NN}=17.3$ AGeV. The extracted
source exhibits long-range non-Gaussian tails in the directions of the pion-
pair net transverse-momentum and of the beam. Comparison with the Therminator
model allows for an extraction of the pion source proper breakup time and of
emission duration in the collisions.

##  1 Introduction

A deconfined phase of nuclear matter is expected to be formed at the high
energy densities created in relativistic heavy ion collisions qgp06 . It is
expected that the signatures of such a phase are reflected in the space-time
extent and shape of particle emission source-functions.

Recently, one-dimensional (1D) source imaging techniques brown97 ; brown98 ;
brown01  have revealed a non-trivial long-range structure in the two-pion
emission source at RHIC chung05 ; chung06 . The origins of this structure are
still investigated. The nature of such structures and the potentially useful
information they can provide could be revealed, on one hand, by examining the
possible presence of such a structure in heavy-ion collisions at intermediate
SPS energies and, on the other, by examining the dependence of a structure on
the direction within the source. The NA49 Collaboration has investigated Pb+Pb
collisions over a wide range of bombarding energies at the CERN SPS during the
last decade alt05 , accumulating pion data. The rich data set provides a
unique opportunity to search for non-trivial structures at the SPS,
investigate their dependence on energy and direction and extract any useful
information those structures may provide.

In this paper, the first three-dimensional (3D) emission source image for low
$p_{T}$ pion pairs produced in central Pb+Pb collisions at
$\sqrt{s}_{NN}=17.3$ GeV is presented. The result is compared with
calculations from the Therminator model in order to extract the source proper
breakup time and emission duration.

##  2 Setup and Analysis

###  2.1 Experimental Setup

The data presented here were taken by the NA49 Collaboration during the years
1996-2002. The incident beams of 158 AGeV were provided by the CERN SPS
accelerator. The NA49 Large Acceptance Hadron Detector afa99  achieves a
large-acceptance precision-tracking ($\delta p/p^{2}\approx(0.3-7).10^{-4}$
(GeV/c$)^{-1}$) and particle identification using time projection chambers
(TPC’s). Charged particles are detected by the tracks left in the TPC’s and
identified by the energy deposited in the TPC gas. Mid-rapidity particle
identification is further enhanced by a time-of-flight wall (resolution of 60
ps). Event centrality is determined by a forward calorimeter which measures
the energy of spectator matter.

###  2.2 Three-Dimensional Correlation Function

The 3D correlation function, C($\mathbf{q}$), was calculated as the ratio of
pair to uncorrelated reference distributions in relative momentum $\mathbf{q}$
for $\pi^{+}\pi^{+}$ and $\pi^{-}\pi^{-}$ pairs. Here,
$\mathbf{q}=\frac{(\mathbf{p_{1}}-\mathbf{p_{2}})}{2}$ is half of the momentum
difference between the two particles in the pair Center-of-Mass System (PCMS)
frame. The pair distribution was obtained using pairs of particles from the
same event and the uncorrelated distribution was obtained by pairing particles
from different events. The Lorentz transformation of $\mathbf{q}$ from the
laboratory frame to the PCMS was done by a Lorentz transformation to the pair
Locally Co-Moving System (LCMS) frame along the beam direction followed by a
Lorentz transformation to the PCMS along the direction of the transverse
momentum of the pair.

Track merging and splitting effects were suppressed by appropriate cuts on
both the pair and uncorrelated distributions. The pair cuts require the two
particles in each pair to be separated by at least 2.2 cm over 50 pad rows in
the vertex TPC’s. A 20$\%$ increase in this minimum separation results in the
correlation data points fluctuating within the statistical errors. Hence, the
systematic uncertainty associated with the pair cuts is deemed smaller than
the statistical uncertainty.

The effects of track momentum resolution were assessed by jittering the
momentum of the tracks in the data by the maximum momentum resolution, $\delta
p/p^{2}\approx 7.10^{-4}$ (GeV/c$)^{-1}$, and by re-computing the 3D
correlation function. The resulting correlation function incorporates then
twice the effect of the momentum resolution and was found to be very close to
the raw un-smeared correlation function. This is not surprising, considering
that the mean momentum of the tracks used in this analysis is 1.4 GeV/c,
resulting in a momentum resolution of $\delta p/p\approx 0.1\%$. The above
procedure gives a very small smear to the correlation peak in the region
$q<10$ MeV. In practice, this tends to reduce the imaged source intensity at
large separation. As a consequence, the results for the source function
presented here may slightly underestimate the actual source function at large
$r$.

###  2.3 Angular Moments

In the Cartesian surface-spherical harmonic decomposition technique daniel05 ;
chung05 , the 3D correlation function is expressed as

$C(\mathbf{q})-1=R(\mathbf{q})=\sum_{l}\sum_{\alpha_{1}\ldots\alpha_{l}}R^{l}_{\alpha_{1}\ldots\alpha_{l}}(q)\,A^{l}_{\alpha_{1}\ldots\alpha_{l}}(\Omega_{\mathbf{q}})$
(1)

where $l=0,1,2,\ldots$, $\alpha_{i}=x,y\mbox{ or }z$,
$A^{l}_{\alpha_{1}\ldots\alpha_{l}}(\Omega_{\mathbf{q}})$ are Cartesian
harmonic basis elements ($\Omega_{\mathbf{q}}$ is solid angle in $\mathbf{q}$
space) and $R^{l}_{\alpha_{1}\ldots\alpha_{l}}(q)$ are Cartesian correlation
moments given by

$R^{l}_{\alpha_{1}\ldots\alpha_{l}}(q)=\frac{(2l+1)!!}{l!}\int\frac{d\Omega_{\mathbf{q}}}{4\pi}A^{l}_{\alpha_{1}\ldots\alpha_{l}}(\Omega_{\mathbf{q}})\,R(\mathbf{q})$
(2)

where $q$ is the modulus of the 4-vector $\mathbf{q}$. The coordinate axes are
oriented so that $z$ is parallel to the beam (long) direction, $x$ points in
the direction of the total momentum of the pair in the locally co-moving
system (LCMS) (out), and $y$ is perpendicular to the first two axes (side).

The correlation moments, for each order $l$, may be calculated from the
measured 3D correlation function using Eq. (2), but can be then vulnerable to
directional inefficiencies. Alternatively, the moments may be fitted using Eq.
(1), avoiding regions of poor efficiency or accounting for those in the fit
weights. In following the latter type of analysis, Eq. (1) has been truncated
at $l=4$ and expressed in terms of independent moments only. The higher order
moments have been found to be negligible. Up to order 4, there are 6
independent moments: $R^{0}$, $R^{2}_{x2}$, $R^{2}_{y2}$, $R^{4}_{x4}$,
$R^{4}_{y4}$ and $R^{4}_{x2y2}$ where $R^{2}_{x2}$ is shorthand for
$R^{2}_{xx}$ etc. These independent moments were extracted as a function of
$q$, by fitting the truncated series to the measured 3D correlation function
with the moments as the parameters of the fit.

###  2.4 Source Reconstruction

Each independent correlation moment can be imaged using the 1D source imaging
code of Brown and Danielewicz brown97 ; brown98 ; brown01  to obtain the
corresponding source moment within each order $l$. Both the effects of Bose-
Einstein symmetrisation and of Coulomb interaction, the sources of the
observed correlations, are accounted for in the source imaging code.
Thereafter, the total source function, $S(\mathbf{r})$, can be reconstructed
by combining the source moments for each $l$:

$S(\mathbf{r})=\sum_{l}\sum_{\alpha_{1}\ldots\alpha_{l}}S^{l}_{\alpha_{1}\ldots\alpha_{l}}(r)\,A^{l}_{\alpha_{1}\ldots\alpha_{l}}(\Omega_{\mathbf{r}})$
(3)

Another method of construction of the 3D source function from the 3D
correlation function is by fitting the latter directly upon assuming the shape
for the source function. Since the 3D correlation function is represented by
the Cartesian moments in the Cartesian harmonic decomposition, this amounts to
fitting the six independent moments, dependent on $q$, with a trial source
function. Two trial functional shapes were considered in this analysis: the 3D
Gaussian, frequently termed an ellipsoid shape, and the Hump function, with
six adjustable parameters, given by

$S(x,y,z)=\lambda\exp[-f_{s}(\frac{r^{2}}{4r_{s}^{2}})-f_{l}(\frac{x^{2}}{4r_{xl}^{2}}+\frac{y^{2}}{4r_{yl}^{2}}+\frac{z^{2}}{4r_{zl}^{2}})]$
(4)

where $f_{s}=1/[1+(r/r_{0})^{2})]$ and $f_{l}=1-f_{s}$.

##  3 Results

Figure 1 (a) shows a comparison between the 1D correlation function C($q$) and
the $l=0$ moment $1+R(q)$ for mid-rapidity low-$p_{T}$ pion-pairs from central
Pb+Pb collisions. The functions are in very good agreement with each other as
expected in the absence of significant detection inefficiencies with
dependence on angle in $\mathbf{q}$-space. The figure also shows that the Hump
function (solid line) fits the data very well while the ellipsoid fit (dotted
line) underestimates the data at $q<13$ MeV/c. This difference in correlation
fits at low $q$ corresponds to a difference in the source function at large
$r\gtrsim 15$ fm, as evident in Fig. 1 (b). The Hump shape (solid line) is in
good agreement with the source image (squares) whereas the ellipsoid shape
(dotted line) underestimates the image. Given that the discrepancy occurs at
large $r$, that discrepancy becomes even more pronounced for the radial
density in Fig. 1(c).

Figure 2 shows the different $l=2$ and $l=4$ moments (open circles) as a
function of pion separation. The $l=4$ moments are significantly smaller in
magnitude compared to the $l=2$ moments: this justifies the truncation of the
series Eq. (1) at $l=4$. The Hump fit (solid line), with
$\chi^{2}/\text{ndf}=1.3$, is in close agreement with the data whereas the
ellipsoid function (dotted line) gives a poor fit to the data
($\chi^{2}/\text{ndf}=6.8$), as is visually evident.

Figure 3 (a)-(c) shows the 3D correlation function profiles in the $x$, $y$
and $z$ directions while the corresponding source function profiles are shown
in panels (d)-(f). The source image (squares) is in good agreement with the
Hump function fit (solid line), but is underestimated by the ellipsoid fit
(dotted line) in all 3 directions. These non-Gaussian tails in the Hump source
function profiles have corresponding manifestations in the correlation
function profiles at low $q$, relative to the ellipsoid fit.

##  4 Therminator Model

###  4.1 Model Assumptions

The Therminator model kis05  can shed more light on the source breakup and
emission dynamics. The Therminator model incorporates the following: (1) the
Bjorken assumption of longitudinal boost invariance, (2) blast-wave expansion
in the transverse direction with transverse velocity profile semilinear in
transverse radius $\rho$ kis07 , (3) thermal emission of particles from a
cylinder of infinite longitudinal size and finite transverse radius
$\rho_{max}$, and (4) all known resonance decays.

Assumptions (1) and (2) of the model imply that the expanding source consists
of fluid elements shaped like onion rings with their axes aligned with the
beam axis. Each fluid element expands transversely with a transverse velocity
semi-linear in $\rho$, as well as translates longitudinally with a velocity
profile linear in $z$.

Each fluid element breaks up after a proper breakup time $\tau$ in its own
rest frame, with subsequent particle emission. Bjorken assumption of
longitudinal boost invariance implies that the breakup (emission) time, $t$,
of a source element at position $z$ in the laboratory frame is given by
$t^{2}=\tau^{2}+z^{2}$ .

At the point of source breakup, particles within the Therminator model leave
each source element defined by their longitudinal and transverse positions:
$z$ and $\rho$. All particle emissions are externally viewed as occuring from
a freeze-out hypersurface defined in $\rho$-$\tau$ plane as
$\tau=\tau_{0}+a\rho$, where $a$ is a parameter which controls the space-time
correlation and $\tau_{0}$ is the proper breakup time of the source element
located at $\rho=0$, i.e. on the beam axis, and $\tau$ is proper breakup time
of the source element located at finite $\rho$. Hence, particles which are
emitted from a generic fluid element with coordinates ($z$,$\rho$) will have
emission time $t$ in the laboratory frame given by
$t^{2}=(\tau_{0}+a\rho)^{2}+z^{2}$ .

In this analysis, Therminator is used in the blast-wave mode and $a$ is set to
the negative value of $-0.5$ kis06 . Hence, $\tau$ decreases linearly with
$\rho$ down to a minimum value at $\rho=\rho_{max}$. The negative value of $a$
implies a negative space-time correlation, i.e. particles at large $\rho$
freeze-out earlier. Hence, the source emits or burns from outside in.

Since each source element is defined by only one value of the proper breakup
time $\tau$, all particle emissions from this fluid element happen
instantaneously in the rest frame of the source element and the proper
emission duration in the above scenario is 0. On the other hand, one can allow
for a finite non-zero proper emission duration in each source element, if the
single proper breakup time $\tau$ of each source element is replaced with a
distribution of breakup times. This corresponds physically to the source
element breaking up over a finite time interval rather than instantaneously.

One such parametrization is an exponential distribution of breakup times
$\tau^{\prime}$ with a width $\Delta\tau$ given by
$dN/d\tau^{\prime}=\frac{\Theta(\tau^{\prime}-\tau)}{\Delta\tau}\,\text{exp}[-(\tau^{\prime}-\tau)/\Delta\tau]$
. The minimum value of $\tau^{\prime}$ is the initial single breakup time
$\tau$ for that source element. With such a distribution of breakup times,
each source element emits particles from a family of hypersurfaces, each of
which is defined by a $\tau^{\prime}$ value which is sampled according to the
exponential distribution. In such a scenario, particle emission from each
source element takes place over a finite time duration defined by $\Delta\tau$
rather than instantaneously. In this parametrization, $\Delta\tau$ represents
the proper emission duration in the rest frame of the fluid element.

###  4.2 Comparison to Data

Figure 4 shows a comparison of the source image (squares) with the calculated
source function (circles) from the Therminator model in its blast-wave mode. A
good match is obtained with values of $\tau_{0}=7.3$ fm/c and $\Delta\tau=3.7$
fm/ c, when all resonance decays are turned on. The good agreement between
extracted source image and calculated source function indicates that the data
are consistent with the basic ingredients of Bjorken longitudinal expansion
and with the blast-wave dynamics for transverse expansion which are
incorporated in the Therminator model. It also indicates that a finite non-
zero pion emission duration and outside-in burning of the source are needed to
describe the data.

##  5 Conclusions

The model-independent 3D source imaging technique, involving decomposition of
the 3D correlation function into Cartesian surface-spherical harmonics, has
been applied to pion pairs from Pb+Pb collisions at $\sqrt{s}_{NN}=17.3$ GeV.
Prominent non-Gaussian tails have been observed in the outward direction of
the pion pair transverse momentum and in the longitudinal direction of the
beam. The extracted source image is well described by the Therminator model
which incorporates Bjorken longitudinal expansion and blast-wave transverse-
flow dynamics. The data is consistent with a proper emission duration of 3.7
fm/c for pions at $\sqrt{s}_{NN}=17.3$ GeV.

## References

  * (1) Ma Y-G, Wang E-K, Cai X, Huang H-Z, Wang X-N and Zhu Z-Y (ed) 2007 J. Phys. G: Nucl. Part. Phys. 34 issue 8 (Proc. QUARK MATTER 2007).
  * (2) D. A. Brown and P. Danielewicz, Phys. Lett. B 398, 252 (1997).
  * (3) D. A. Brown and P. Danielewicz, Phys. Rev. C 57, 2474 (1998).
  * (4) D. A. Brown and P. Danielewicz, Phys. Rev. C 64, 14902 (2001).
  * (5) P. Chung et al., Nucl. Phys. A 749, 275c (2005).
  * (6) S. S. Adler et al. (PHENIX Collaboration), Phys. Rev. Lett. 98, 132301 (2007)
  * (7) C. Alt et al. (NA49 Collaboration), CERN-SPSC-2005-041, CERN-SPSC-P-264-ADD-12, Nov 2005.
  * (8) S. Afanasiev et al., Nucl. Instrum. Meth. A430 210 (1999).
  * (9) P. Danielewicz and S. Pratt, Phys. Lett. B 618 60 (2005).
  * (10) A. Kisiel et al., Comput. Phys. Commun. 174, 669, 2006.
  * (11) A. Kisiel, Braz. J. Phys. 37, 3A, 917 (2007).
  * (12) A. Kisiel et al., Phys. Rev. C 73, 064902 (2006).

5mm

Figure 1:  (color on line) Angle averaged correlation function (top panel) and
source function (middle) and radial probability density (bottom). Filled
circles show correlations from a direct angle-averaging of the data. Open
circles represent correlations from fitting the angular decomposition to the
data. Squares show the imaged source and correlations corresponding to the
imaged source. The dotted and solid lines represent, respectively, the fitted
Gaussian and Hump sources and their corresponding correlation functions (see
text).

5mm

Figure 2:  (color online) Correlation moments of multi-polarity $l=2$ (left
panels), and $l=4$ (right panels) for $\pi^{+}\pi^{+}$ and $\pi^{-}\pi^{-}$
pairs. Results are represented in the same way as in Fig. 1, with error bars
indicating statistical errors for the data. Systematic errors are smaller than
the data points.

5mm

Figure 3:  (color online) Correlation $C(q)$ (left panels) and source $S(r)$
(right panels) function profiles for $\pi^{+}\pi^{+}$ and $\pi^{-}\pi^{-}$
pairs in the outward $x$ (top panels), sideward $y$ (middle) and longitudinal
$z$ (bottom) directions. The use of symbols is analogous to that in Fig. 1.
Error bars indicate statistical errors. Systematic errors do not exceed the
symbol size. Here, $l=4$ moments make negligible contributions.

5mm

Figure 4:  Source function profiles, $S(r_{i})$, along $x$ (a), $y$ (b) and
$z$ (c) directions, from imaged correlation function (squares) and from the
Therminator model (circles).


266282

# Foliations with all non-closed leaves on non-compact surfaces

Sergiy Maksymenko  Institute of Mathematics, National Academy of Sciences of
Ukraine, 3 Tereshchenkivs’ka, Kyiv, 01601, Ukraine  Sergiy Maksymenko
<EMAIL_ADDRESS>Eugene Polulyakh  Institute of Mathematics, National
Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka, Kyiv, 01601, Ukraine
Eugene Polulyakh<EMAIL_ADDRESS>
(Date: 30/05/2016)

###### Abstract.

Let $X$ be a connected non-compact $2$-dimensional manifold possibly with
boundary and $\Delta$ be a foliation on $X$ such that each leaf
$\omega\in\Delta$ is homeomorphic to $\mathbb{R}$ and has a trivially foliated
neighborhood. Such foliations on the plane were studied by W. Kaplan who also
gave their topological classification. He proved that the plane splits into a
family of open strips foliated by parallel lines and glued along some boundary
intervals. However W. Kaplan’s construction depends on a choice of those
intervals, and a foliation is described in a non-unique way. We propose a
canonical cutting by open strips which gives a uniqueness of classifying
invariant. We also describe topological types of closures of those strips
under additional assumptions on $\Delta$.

###### Key words and phrases:

Foliation, non-compact surface, fiber bundles.

###### 2010 Mathematics Subject Classification:

57R30, 55R10

##  1\. Introduction

Let $X$ be a $2$-dimensional manifold possibly non-connected and having a
boundary, and $\Delta$ be a one-dimensional foliation on $X$. We will say that
$\Delta$ belongs to class $\mathcal{F}$ if it satisfies the following three
conditions.

  1. (1)

Each leaf $\omega$ of $\Delta$ is a closed subset of $X$.

  2. (2)

Every connected component $\omega$ of $\partial X$ is a leaf of $\Delta$.

  3. (3)

Let $\omega\in\Delta$ be a leaf, and $J=[0,1)$ if $\omega\subset\partial X$,
and $J=(-1,1)$ otherwise. Then there exists an open neighborhood $U$ of
$\omega$ and a homeomorphism $\phi:\mathbb{R}\times J\to U$ such that
$\phi(\mathbb{R}\times 0)=\omega$ and $\phi(\mathbb{R}\times t)$ is a leaf of
$\Delta$ for all $t\in J$, see Figure 1.1.

Roughly speaking, _a $1$-dimensional foliation_ $\Delta$ is a partition of $X$
which looks like a partition of $\mathbb{R}^{2}$ into parallel lines _near
each point $x\in X$_. Then $\Delta$ belongs to class $\mathcal{F}$ whenever it
looks like partition of $\mathbb{R}^{2}$ into parallel lines _near each leaf
$\omega\in\Delta$_. In particular, each leaf of $\Delta$ is homeomorphic to
$\mathbb{R}$.

|  |
---|---|---
(a) leaf in the boundary |  | (b) leaf in the interior
Figure 1.1.

######  Definition 1.1.

Let $X_{i}$ be a surface with a foliation $\Delta_{i}$, $i=1,2$. Then a
homeomorphism $h:X_{1}\to X_{2}$ will be called _foliated_ if it maps leaves
of $\Delta_{1}$ onto leaves of $\Delta_{2}$. In this case we will also write
$h:(X_{1},\Delta_{1})\to(X_{2},\Delta_{2})$.

The aim of the present paper is to describe a topological structure of
foliations belonging to class $\mathcal{F}$ up to foliated homeomorphisms, see
Theorem 1.8 below. Such foliations on the plane were studied by W. Kaplan [13]
and they appear as foliations by level sets of pseudoharmonic functions on
$\mathbb{R}^{2}$, see W. Kaplan [13, Theorem 42], W. Boothby [6], [7], M.
Morse and J. Jenkins [12], M. Morse [16]. We will improve Kaplan’s
construction and extend it to foliations on arbitrary surfaces.

Topological structure of singular foliations on surfaces, in particular,
foliations by orbits of flows, were studied by A. Andronov and L. Pontryagin
[1], M. Peixoto [18], [19], S. Aranson and V. Grines [2, 3], I. Bronstein and
I. Nikolayev [8], S. Aranson, E. Zhuzhoma, and V. Medvedev [5], L. Plachta
[22, 20, 21], A. Oshemkov and V. Sharko [17], S. Aranson, V. Grines and V.
Kaimanovich [4], M. Farber [11], N. Budnytska and O. Prishlyak [9], N.
Budnyts’ka and T. Rybalkina [10] and many others. Results of the paper could
also be applied to singular foliations without non-closed leaves on surfaces
by removing singularities. This will be done in subsequent papers by the
authors.

### Special leaves

Suppose $\Delta$ is a foliation of class $\mathcal{F}$ on a surface $X$. Let
$Y=X/\Delta$ be the space of leaves, and $p:X\to Y$ be the corresponding
quotient map. Endow $Y$ with the _quotient topology_ , so a subset $V\subset
Y$ is open if and only if its inverse $p^{-1}(V)$ is open in $X$. For a subset
$U\subset X$ its _saturation_ , $S(U)$, with respect to $\Delta$ is the union
of all leaves of $\Delta$ intersecting $U$. Equivalently, $S(U)=p^{-1}(p(U))$.

Since each leaf of $\Delta$ is a closed subset of $X$, it follows that $Y$ is
a $T_{1}$-space. However, in general, $Y$ is not a Hausdorff space.

###### Lemma 1.2

If $\Delta\in\mathcal{F}$ then the projection map $p:X\to Y$ is open.

###### Proof.

We have to prove that for each open $V\subset X$ its saturation $S(V)$ is open
as well. Thus for each $x\in S(V)$ we should find an open saturated subset $W$
such that $x\in W=S(W)\subset S(V)$. Let $\omega$ be the leaf containing $x$.
Put $J=[0,1)$ whenever $\omega\subset\partial X$ and $J=(-1,1)$ otherwise.
Then by definition of class $\mathcal{F}$ there exists a foliated
homeomorphism $\phi:\mathbb{R}\times J\to U$ such that
$\phi^{-1}(x)=(t,0)\in\mathbb{R}\times 0$ for some $t\in\mathbb{R}$. Then
$\phi^{-1}(V\cap U)$ is an open neighborhood of $(t,0)$, whence there exists
$\varepsilon>0$ such that if we denote $K=J\cap(-\varepsilon,\varepsilon)$,
then $t\times K\subset\phi^{-1}(V\cap U)$. But $K$ is open in $J$, whence
$\mathbb{R}\times K$ is open in $\mathbb{R}\times J$. Therefore
$\phi(\mathbb{R}\times K)$ is saturated and open in $U$ which in turn is open
in $X$. Hence $\phi(\mathbb{R}\times K)$ is open in $X$ and
$x\in\phi(\mathbb{R}\times K)\subset S(V)$. Therefore $S(V)$ is open in $X$. ∎

######  Definition 1.3.

Let $\omega$ be a leaf of $\Delta$ and $y=p(\omega)\in Y$. We will say that
$\omega$ is a _special_ leaf and $y$ is a _special_ point of $Y$ whenever $Y$
is not Hausdorff at $y$, that is $y\not=\cap_{y\in V}\overline{V}$, where $V$
runs over all open neighborhoods of $y$.

######  Example 1.4.

Consider the foliation on $\mathbb{R}^{2}$ shown in Figure 1.2(a). It splits
by bold leaves $\alpha$, $\beta$, $\gamma$, and $\delta$ into five “strips”
$A$, $B$, $C$, $D$, $E$ foliated by “parallel” lines, see Figure 1.2(b).
Moreover, the space of leaves $Y$ has the structure as in Figure 1.2(c), where
bold lines correspond to strips, and thin lines just indicate that $\alpha$
belongs to the closure of $A$ and $B$, $\beta$ belongs to the closures $B$ and
$C$ and so on. In particular, $Y$ looses Hausdorff property at $\alpha$,
$\beta$, $\gamma$, and $\delta$. More precisely, the subspace
$Y\setminus\\{\alpha,\beta,\gamma,\delta\\}$ is Hausdorff, however each
neighborhood of $\alpha$ intersect each neighborhood of $\beta$, and the same
holds for pairs $\\{\beta,\gamma\\}$ and $\\{\gamma,\delta\\}$. Therefore the
leaves $\alpha$, $\beta$, $\gamma$ and $\delta$ are special.

|  |  |  |
---|---|---|---|---
(a) Foliation $\Delta$ |  | (b) Strips decomposition |  | (c) Space of leaves $Y=X/\Delta$
Figure 1.2.

######  Definition 1.5.

A subset $S\subset\mathbb{R}^{2}$ will be called a _model strip_ if there
exist $a<b$ such that

  1. (1)

$\mathbb{R}\times(a,b)\ \subset\ S\ \subset\ \mathbb{R}\times[a,b]$;

  2. (2)

the intersection $S\ \cap\ \mathbb{R}\times\\{a,b\\}$ is a disjoint union of
open intervals.

Put

$\displaystyle\partial_{-}S$ $\displaystyle=S\cap(\mathbb{R}\times\\{a\\}),$
$\displaystyle\partial_{+}S$ $\displaystyle=S\cap(\mathbb{R}\times\\{b\\}),$
$\displaystyle\partial S$ $\displaystyle=\partial_{-}S\cup\partial_{+}S.$

A model strip $\mathbb{R}\times(a,b)$ will be called _open_.

Each model strip $S$ admits a natural $1$-dimensional foliation into parallel
lines $\mathbb{R}\times t$ and boundary intervals from $\partial S$. We will
call this foliation _canonical_. The following lemma implies that this
foliation belongs to class $\mathcal{F}$.

###### Lemma 1.6

(e.g. [15]). Let $a<b\in\mathbb{R}$,
$X=\mathbb{R}^{2}\setminus\bigl{(}(-\infty,a]\cup[b,+\infty)\bigr{)}$, and
$\varepsilon>0$. Then there exists a homeomorphism $\phi:\mathbb{R}^{2}\to X$
such that

  1. (a)

$\phi$ is fixed outside $\mathbb{R}\times(-\varepsilon,\varepsilon)$;

  2. (b)

$\phi$ preserves foliations by horizontal lines, that is
$\phi(\mathbb{R}\times t)=t\times\mathbb{R}$ for $t\not=0$ and
$\phi(\mathbb{R}\times 0)=(a,b)\times 0$, see Figure 1.3.

Figure 1.3.

######  Example 1.7.

The foliation in Example 1.4 splits into five model strips such that

$\displaystyle A\ \cong\ E\ \cong\ \mathbb{R}\times(0,1)\ \bigcup\ (0,1)\times
1,$ $\displaystyle B\ \cong\ C\ \cong\ D\ \cong\ \mathbb{R}\times(0,1)\
\bigcup\ \bigl{(}(0,1)\cup(2,3)\bigr{)}\times 1.$

Let $\mathbb{R}\times[-1,1]$ be a model strip,
$\phi_{+},\phi_{-}:\mathbb{R}\times\\{-1\\}\to\mathbb{R}\times\\{+1\\}$ be two
homeomorphisms given by

$\displaystyle\phi_{+}(t,-1)$ $\displaystyle=(t,1),$
$\displaystyle\phi_{-}(t,-1)$ $\displaystyle=(-t,1),$

for $t\in\mathbb{R}$, and $C=\mathbb{R}\times[-1,1]/\phi_{+}$ and
$M=\mathbb{R}\times[-1,1]/\phi_{-}$ be the quotient spaces. Thus $C$ (resp.
$M$) is obtained from $\mathbb{R}\times[-1,1]$ by identifying its boundary
lines via preserving (resp. reversing) orientation homeomorphism. Therefore
$C$ is a cylinder and $M$ is a Möbius band. Moreover, the canonical foliation
on $\mathbb{R}\times[-1,1]$ yields certain foliations $\Delta_{C}$ and
$\Delta_{M}$ on $C$ and $M$ respectively also belonging to class
$\mathcal{F}$. We will call $C$ a _standard cylinder_ and $M$ a _standard
Möbius band_.

### Foliation associated with a regular function.

A continuous function $f:\mathbb{R}^{2}\to\mathbb{R}$ will be called _regular_
whenever for each $z\in\mathbb{R}^{2}$ there are local coordinates $(u,v)$ in
which $z=(0,0)$ and $f(u,v)=u+\mathrm{const}$.

It follows that the partition $\Delta$ of $\mathbb{R}^{2}$ into connected
components of level-sets $f^{-1}(t)$, $t\in\mathbb{R}$, of $f$ is a
_foliation_ is a usual sense, i.e. it is _locally_ homeomorphic with a
partition of $\mathbb{R}^{2}$ into parallel lines. We will say that $\Delta$
is a _foliation associated with $f$_.

Notice that _$f$ has no local extremes_, whence all leaves of $f$ are
homeomorphic with $\mathbb{R}$. Indeed, if $\Delta$ has a closed leaf
$\omega$, then by Jordan theorem $\omega$ bounds a $2$-disk. Since $f$ is
constant on $\omega$, it must have a local extreme inside that disk, which
gives a contradiction.

Let $J\subset\mathbb{R}$ be a connected subset, i.e. either open or closed or
half-closed interval. Then by a _cross-section_ $\sigma:J\to\mathbb{R}^{2}$ of
$\Delta$ we will mean a continuous path intersecting each leaf at most once.
It easily follows that $\sigma$ is a cross-section if and only if the
composition $f\circ\sigma:J\to\mathbb{R}$ is strictly monotone.

By a _saturation of a cross-section_ $\sigma:J\to\mathbb{R}^{2}$ we will mean
the saturation of its image $S(\sigma(J))$ and denote it simply by
$S(\sigma)$, c.f. [13, §1.4]

Kaplan [13, Theorem 30] proved that for a cross-section
$\sigma:[a,b]\to\mathbb{R}^{2}$ of $\Delta$ its saturation $S(\sigma)$ is
foliated homeomorphic with $\mathbb{R}\times[a,b]$ foliated by parallel lines.
However, this result can be misleading, since $S(\sigma)$ is not necessarily a
closed subset of $X$.

For instance, consider the foliation in Figure 1.2(b). Let
$\sigma:[a,b]\to\mathbb{R}^{2}$ be a cross-section passing through the special
leaf $\alpha$ and such that $\sigma(a)\in A$ and $\sigma(b)\in B$. Then
$\overline{S(\sigma)}\setminus S(\sigma)=\beta$.

### Kaplan’s construction.

In [13, Theorem 29] W. Kaplan has shown that the foliation $\Delta$ associated
with a regular function $f$ belongs to class $\mathcal{F}$. In fact, he
associated to $\Delta$ a family of pairs
$\xi=\\{(\omega_{i},\sigma_{i})\\}_{i=-a}^{b}$ for some
$a,b\in\mathbb{N}\cup\\{\infty\\}$, where

  * (i)

$\omega_{i}$ is a leaf being special for $i\not=0$;

  * (ii)

$\sigma_{0}:(-1,1)\to\mathbb{R}^{2}$, $\sigma_{i}:[0,1)\to\mathbb{R}^{2}$ for
$i>0$, and $\sigma_{i}:(-1,0]\to\mathbb{R}^{2}$ for $i<0$ are certain proper
cross-sections of $\Delta$;

  * (iii)

$\sigma_{i}(0)\in\omega_{i}$ for all $i$,

$\displaystyle\sigma_{i}[0,1)\ \cap\
S\Bigl{(}\mathop{\cup}\limits_{j=0}^{i-1}\sigma_{j}[0,1)\Bigr{)}$
$\displaystyle=\sigma_{i}(0),$ $\displaystyle i>0,$
$\displaystyle\sigma_{i}(-1,0]\ \cap\
S\Bigl{(}\mathop{\cup}\limits_{j=i+1}^{0}\sigma_{j}(-1,0]\Bigr{)}$
$\displaystyle=\sigma_{i}(0),$ $\displaystyle i<0.$

Kaplan proved that $\xi$ determines $\Delta$ up to a foliated homeomorphism.

As noted above $S(\sigma_{0})$ is foliated homeomorphic with
$\mathbb{R}\times(0,1)$ while $S(\sigma_{i})$, $i\not=0$, is foliated
homeomorphic with a strip $\mathbb{R}\times[0,1)$. Therefore the family $\xi$
determines at most countable family of strips $\\{V_{i}=S(\sigma_{i})\\}$ such
that $V_{i+1}$ is glued to $V_{i}$ along the interval $\omega_{i}$ in their
boundaries.

Kaplan’s aim was to decrease the family of such strips as much as possible,
see first paragraph of [13, Section 3.1]. However, the construction of family
$\xi$ then becomes ambiguous and depends on a particular choice of special
leaves and cross-sections. This is illustrated in Figure 1.4(b), where two
such families for the same foliation are presented.

|  |
---|---|---
(a) Foliation |  | (b) Two distinct maximal families of cross-sections
Figure 1.4.

On the other hand cutting $\mathbb{R}^{2}$ along special leaves is an
unambiguous procedure and it gives a canonical decomposition of
$\mathbb{R}^{2}$.

In the present paper we extends Kaplan’s results to foliations $\Delta$ from
class $\mathcal{F}$ on arbitrary surfaces $X$ and describe the topological
structure of connected components of $X\setminus\Sigma$ and their closures,
where $\Sigma$ is the union of all special leaves of $\Delta$.

###### Theorem 1.8

Let $X$ be a connected $2$-dimensional manifold and $\Delta$ be a foliation on
$X$ belonging to class $\mathcal{F}$. Suppose that the family $\Sigma$ of all
special leaves of $\Delta$ is locally finite, and let $Q$ be a connected
component of $X\setminus(\Sigma\cup\partial X)$. Then the following statements
hold true.

  1. (1)

$Q$ is foliated homeomorphic either with a standard cylinder $C$ or a standard
Möbius band $M$ or an open model strip $\mathbb{R}\times(-1,1)$. Moreover, in
the first two cases $Q=X$.

  2. (2)

Suppose $Q$ is foliated homeomorphic with an open model strip. Fix any
foliated homeomorphism $\phi:\mathbb{R}\times(-1,1)\to Q$ and denote

$\displaystyle A$ $\displaystyle=\phi\bigl{(}\mathbb{R}\times(-1,0]\bigr{)},$
$\displaystyle B$ $\displaystyle=\phi\bigl{(}\mathbb{R}\times[0,1)\bigr{)}.$

Then the closures $\overline{A}$ and $\overline{B}$ are foliated homeomorphic
to some model strips.

This theorem implies that the topological structure of the foliation
$\Delta\in\mathcal{F}$ is uniquely determined by the combinatorics of gluing
model strips. Also notice that the intersection
$(\overline{A}\cap\overline{B})\setminus\phi(\mathbb{R}\times 0)$ can be non-
empty, whence one can not expect that
$\overline{Q}=\overline{A}\cup\overline{B}$ is homeomorphic with a model
strip.

The proof of Theorem 1.8 will be given in §4 and 6.

##  2\. Special points of non-Hausdorff spaces

Throughout this section $Y$ be a topological space.

######  Definition 2.1.

Let $y\in Y$ and $\beta_{y}$ be the family of all neighborhoods of $y$. Then
the following set

$\mathrm{hcl}(y):=\mathop{\cap}\limits_{V\in\beta_{y}}\overline{V}$

will be called the _Hausdorff closure_ of $y$. We will say that $y$ is a
_special_ point of $Y$ whenever $y\not=\mathrm{hcl}(y)$. The set of all
special points of $Y$ will be denoted by $\mathcal{V}$.

Notice that $Y$ is Hausdorff if and only if $y=\mathrm{hcl}(y)$ for all $y\in
Y$, i.e. when $\mathcal{V}=\varnothing$.

###### Lemma 2.2

  1. (1)

Let $y,z\in Y$. Then $y\in\mathrm{hcl}(z)$ if and only if
$z\in\mathrm{hcl}(y)$, however, in general,
$\mathrm{hcl}(y)\not=\mathrm{hcl}(z)$.

  2. (2)

Let $f:Y\to Z$ be a continuous map into a Hausdorff topological space $Z$.
Then $f(\mathrm{hcl}(y))=f(y)$ for all $y\in Y$.

  3. (3)

The set $Y\setminus\mathcal{V}$ of all non-special points is Hausdorff.

###### Proof.

(1) Suppose
$y\in\mathrm{hcl}(z)=\mathop{\cap}\limits_{V\in\beta_{z}}\overline{V}$, that
is $y$ belongs to the closure of each neighborhood of $z$ which means in turn
that every neighborhood of $y$ intersect every neighborhood of $z$. The latter
property is symmetric with respect to $y$ and $z$, whence
$z\in\mathrm{hcl}(y)$ as well.

(2) Suppose $z\in\mathrm{hcl}(y)$ but $f(y)\not=f(z)$. Since $Z$ is Hausdorff,
there exist open disjoint neighborhoods $W_{f(y)}$ and $W_{f(z)}$ of points
$f(y)$ and $f(z)$. But then their inverses $V_{y}=f^{-1}(W_{f(y)})$ and
$V_{z}=f^{-1}(W_{f(z)})$ are disjoint open neighborhoods of $y$ and $z$
respectively. Hence $z$ can not belongs to $\mathrm{hcl}(y)$ which contradicts
to the assumption.

(3) Let $y,z\in Y\setminus\mathcal{V}$ be two distinct points. Thus
$y=\mathrm{hcl}(y)\not=z$ and so there exist disjoint neighborhoods $V_{y}$
and $V_{z}$ of $y$ and $z$ respectively. This implies that
$Y\setminus\mathcal{V}$ is Hausdorff. ∎

### Non-Hausdorff one-dimensional manifolds

Let $Y$ be a $T_{1}$-topological space locally homeomorphic with open sets of
$[0,1)$. Notice that we allow $Y$ to be non-Hausdorff. Then as usual the set
of points having an open neighborhood homeomorphic with $(0,1)$ will be
denoted by $\mathop{\mathrm{Int}}\nolimits{Y}$ and called the _interior_ of
$Y$, while its complement $\partial
Y:=Y\setminus\mathop{\mathrm{Int}}\nolimits{Y}$ will be called the _boundary_
of $Y$.

###### Lemma 2.3

Suppose that the set $\mathcal{V}$ of special points of $Y$ is locally finite.
Then every connected component $W$ of $Y\setminus\mathcal{V}$ is open in $Y$
and is homeomorphic with one of the following spaces: $[0,1)$, $(0,1)$,
$[0,1]$, $S^{1}$. In the last two cases, i.e. when $W$ is compact, $W$ is a
connected component of $Y$.

Every connected component of $Y\setminus(\mathcal{V}\cup\partial Y)$ is
homeomorphic with $(0,1)$.

###### Proof.

Since $Y$ is a $T_{1}$-space, every point $y\in Y$ is a closed subset. Also
since $\mathcal{V}$ is locally finite, it follows that $\mathcal{V}$ is a
closed subset, whence by (3) of Lemma 2.2 $Y\setminus\mathcal{V}$ is a
Hausdorff topological space locally homeomorphic with $[0,1)$. Hence every
connected component $W$ of $Y\setminus\mathcal{V}$ is a one-dimensional
manifold and so it is homeomorphic with one of the spaces $[0,1)$, $(0,1)$,
$[0,1]$, $S^{1}$. Moreover, since $Y\setminus\mathcal{V}$ is locally
connected, we obtain that $W$ is open in $Y\setminus\mathcal{V}$ and therefore
in $Y$ as well.

Suppose $W$ is compact, i.e. it is homeomorphic either with $[0,1]$ or with
$S^{1}$. Let us show that then $W$ is also closed in $Y$. This will imply that
$W$ is a connected component of $Y$.

Let $\\{y_{i}\\}_{i\in\mathbb{N}}\subset W$ be a sequence converging to some
$z\in Y$. We should prove that $z\in W$. Since $W$ is compact, that sequence
also converges to some $y\in W$. Hence if $V_{y}$ and $V_{z}$ are any two open
neighborhoods of $y$ and $z$ respectively, then there exists $n>0$ such that
$y_{n}\in V_{y}\cap V_{z}$. Thus $V_{y}\cap V_{z}\not=\varnothing$, which
implies that $z\in\mathrm{hcl}(y)=\\{y\\}$, that is $z=y\in W$.

We leave the last statement for the reader. ∎

Suppose $Y$ is connected and not homeomorphic with a circle. Let
$\\{W_{\alpha}\\}_{\alpha\in\mathcal{A}}$ be the family of all connected
components of $\mathcal{V}\cup\partial Y$. Then due to Lemma 2.3 for each
$\alpha\in\mathcal{A}$ there exists a homeomorphism $\phi_{\alpha}:(-1,1)\to
W_{\alpha}$. Consider the following collection of subsets:

$\displaystyle\mathcal{B}$
$\displaystyle=\bigl{\\{}\phi_{\alpha}(-1,-\tfrac{1}{2}],\
\phi_{\alpha}[\tfrac{1}{2},1)\bigr{\\}}_{\alpha\in\mathcal{A}}\,.$

Let $K\in\mathcal{B}$. Then we denote $K^{o}:=\phi_{\alpha}(-1,-\tfrac{1}{2})$
if $K=\phi_{\alpha}(-1,-\tfrac{1}{2}]$, and
$K^{o}:=\phi_{\alpha}(\tfrac{1}{2},1)$ if $K=\phi_{\alpha}[\tfrac{1}{2},1)$
for some $\alpha\in\mathcal{A}$. Thus each $K\in\mathcal{B}$ is homeomorphic
with a half-open segment $[0,1)$, and $K^{o}$ is the subset of $K$
corresponding to $(0,1)$.

###### Lemma 2.4

Let $y\in\partial Y$. Then there exists a unique $K\in\mathcal{B}$ such that
$y\in\overline{K}$. In this case $V:=\\{y\\}\cup K^{o}$ is an open
neighborhood of $y$ and there exists a homeomorphism $\psi:[0,1)\to\\{y\\}\cup
K$ such that $\psi(0)=y$ and $\psi(0,1)=V$.

Suppose $y\in\mathcal{V}\setminus\partial Y$. Then there exist two distinct
elements $K,L\in\mathcal{B}$ such that $y\in\overline{K}\cap\overline{L}$ and
$y\not\in\overline{M}$ for all other $M\in\mathcal{B}$. Moreover, the set
$V:=K^{o}\cup\\{y\\}\cup L^{o}$ is an open neighborhood of $y$ and there
exists a homeomorphism $\mu:[-1,1]\to K\cup\\{y\\}\cup L$ such that
$\psi(0)=y$ and $\psi(-1,1)=V$.

###### Proof.

We will consider only the case $y\in\mathcal{V}\setminus\partial Y$. Notice
that the family $\\{\phi_{\alpha}[-1,1]\\}_{\alpha\in\mathcal{A}}$ is locally
finite and consists of closed sets. Therefore its union
$Z=\mathop{\cup}\limits_{\alpha\in\mathcal{A}}\phi_{\alpha}[-1,1]$ is closed.
Hence the set $T=(\mathcal{V}\setminus\\{y\\})\cup\partial Y\cup Z$ is closed
and does not contain $y$. Therefore there exits a neighborhood $J\subset
Y\setminus T$ of $y$ and a homeomorphism $\mu:[-\tfrac{1}{2},\tfrac{1}{2}]\to
J$ such that $\mu(0)=y$ and $\mu(-\tfrac{1}{2},\tfrac{1}{2})$ is an open
neighborhood of $y$.

Notice that $J\setminus\\{y\\}$ consists of exactly two connected components
$A=\mu[-\tfrac{1}{2},0)$ and $B=\mu(0,\tfrac{1}{2}]$, and is contained in
$Y\setminus\bigl{(}\mathcal{V}\cup\partial Y\cup
Z\bigr{)}=\mathop{\cup}\limits_{K\in\mathcal{B}}K^{o}$. Hence $A\subset K^{o}$
and $B\subset L^{o}$ for some $K,L\in\mathcal{B}$, see Figure 2.1.

Figure 2.1.

Moreover, any other neighborhood of $y$ intersects both $A$ and $B$ and
therefore both $K$ and $L$. Hence $y\in\overline{K}\cap\overline{L}$ and
$y\not\in\overline{M}$ for all other $M\in\mathcal{B}$ distinct from $K$ and
$L$.

Fix any homeomorphisms $\kappa:[-1,0)\to K$ and $\lambda:(0,1]\to L$. Notice
that $A$ is not contained in any compact subset $P$ of $K$, since otherwise
$y\in\overline{A}\subset P\subset K$, which contradicts to the assumption that
$y\not\in K$. This implies that $\kappa^{-1}(A)=[a,0)\subset(-1,0)$, where
$a=\kappa^{-1}\circ\mu(-\tfrac{1}{2})\in(-1,0)$. By the same arguments,
$\lambda^{-1}(B)=(0,b]\subset(0,1)$, where
$b=\lambda^{-1}\circ\mu(\tfrac{1}{2})\in(0,1)$.

###### Lemma 2.4.1

$K\not=L$.

###### Proof.

If $K=L$, then we have a homeomorphism
$\mu=\lambda^{-1}\circ\kappa:[-1,0)\to(0,1]$. Hence there exists $c\in(a,0)$
such that $c^{\prime}=\mu(c)\in(0,b)$. Then $\kappa(c)=A$ and
$\lambda\circ\mu(c)\in B$. But $\lambda\circ\mu(c)=\kappa(c)$, and so $A\cap
B\not=\varnothing$, which contradicts to the assumption. ∎

Now fix arbitrary orientation preserving homeomorphisms
$\eta_{K}:[-1,-\tfrac{1}{2}]\to[-1,a]$ and $\eta_{L}:[\tfrac{1}{2},1]\to[b,1]$
and define the map $\psi:[-1,1]\to K\cup\\{y\\}\cup L$ by the formula

$\psi(t)=\begin{cases}\kappa^{-1}\circ\eta_{K}(t),&t\in[-1,-\tfrac{1}{2}],\\\
\mu(t),&t\in[-\tfrac{1}{2},\tfrac{1}{2}],\\\
\lambda^{-1}\circ\eta_{L}(t),&t\in[\tfrac{1}{2},1].\end{cases}$

One easily checks that $\psi$ is a required homeomorphism. ∎

##  3\. Partitions

Let $X$ be a topological space, $\Delta$ be a partition of $X$, $Y=X/\Delta$
be the quotient space, and $p:X\to Y$ be the corresponding quotient map. We
will endow $Y$ with the _factor topology_ , so a subset $V\subset Y$ is open
if and only if its inverse $p^{-1}(V)$ is open in $X$.

A _saturation_ $S(U)$ of a subset $U\subset X$ with respect to $\Delta$ is the
union of all $\omega\in\Delta$ such that $\omega\cap U\not=\varnothing$.
Equivalently, $S(U)=p^{-1}(p(U))$. A subset $U$ is _saturated_ if $U=S(U)$.
Evidently, if $A\cap S(B)=\varnothing$, then $S(A)\cap S(B)=\varnothing$ as
well.

###### Lemma 3.1

  1. (1)

$Y$ is a $T_{1}$-space if and only if each element $\omega\in\Delta$ is
closed.

  2. (2)

The following conditions are equivalent:

    1. (a)

the map $p:X\to Y$ is open;

    2. (b)

for each $x\in X$ there exists an open neighborhood $U$ whose saturation
$S(U)$ is open;

    3. (c)

there exists an open cover $\beta=\\{U_{i}\\}_{i\in\Lambda}$ of $X$ such that
for each $i\in\Lambda$ the restriction $p|_{U_{i}}:U_{i}\to p(U_{i})$ is an
open map.

  3. (3)

If $p$ is open then for each saturated subset $B$ we have that

$\displaystyle X\setminus\overline{B}=S(X\setminus\overline{B}),$ (3.1)
$\displaystyle p(\overline{B})=\overline{p(B)}.$ (3.2)

In particular, $\overline{S(A)}$ and $X\setminus\overline{S(A)}$ are saturated
for each subset $A\subset X$.

  4. (4)

Let $\beta=\\{W_{i}\\}_{i\in\Lambda}$ be a family of subsets of $Y$, and
$\alpha=\\{p^{-1}(W_{i})\\}_{i\in\Lambda}$ be the corresponding family of
their inverses in $X$. If $\beta$ is locally finite, then so is $\alpha$.
Conversely, if $\alpha$ is locally finite and $p$ is open then $\beta$ is
locally finite as well.

  5. (5)

Suppose $X$ is a normal topological space and
$\alpha=\\{\omega_{i}\\}_{i\in\mathbb{N}}$ is a locally finite family of
mutually disjoint closed subsets of $X$. Then for each $i\in\mathbb{N}$ there
exists a neighborhood $U_{i}$ of $\omega_{i}$ such that
$\overline{U_{i}}\cap\overline{U_{j}}=\varnothing$ for $i\not=j$.

  6. (6)

Let $f:A\to B$ be a bijection between topological spaces. Suppose that
$\\{K_{i}\\}_{i\in\Lambda}$ is a locally finite cover of $A$ by closed sets.
If each of the restrictions $f|_{K_{i}}:K_{i}\to B$ is continuous, then $f$ is
continuous it self.

Moreover, suppose the family $\\{\psi(K_{i})\\}_{i\in\Lambda}$ is locally
finite, $f(K_{i})$ is closed in $B$, and the restriction $f|_{K_{i}}:K_{i}\to
f(K_{i})$ is a homeomorphism for each $i\in\Lambda$. Then $f$ is a
homeomorphism.

###### Proof.

Statements (1), (2), and (6) are easy and we leave them for the reader.

(3) Suppose $p$ is an open map and let $B\subset X$ be a saturated subset.
Then $X\setminus B$ is also saturated, i.e. $S(X\setminus B)=X\setminus B$,
and so

$X\setminus\overline{B}\ \subset\ S(X\setminus\overline{B})\ \subset\
S(X\setminus B)=X\setminus B.$

Hence

$\overline{B}\ \supset\ X\setminus S(X\setminus\overline{B})\ \supset\ B.$

As $X\setminus\overline{B}$ is open, $S(X\setminus\overline{B})$ is open as
well, and therefore $X\setminus S(X\setminus\overline{B})$ is a closed subset
containing $B$. Therefore it must contain the closure $\overline{B}$, hence
$\overline{B}=X\setminus S(X\setminus\overline{B})$, which implies (3.1).

Let us prove (3.2). Since $p$ is continuous, $p^{-1}(\overline{p(B)})$ is a
closed subset containing $B$. Therefore it contains $\overline{B}$, and so
$p(\overline{B})\subset\overline{p(B)}$.

Conversely, by (3.1), $\overline{B}$ is saturated and closed. Therefore, by
definition of the quotient topology, $p(\overline{B})$ is a closed subset and
it contains $p(B)$. Hence it also contains $\overline{p(B)}$, i.e.
$p(\overline{B})\supset\overline{p(B)}$.

(4) Suppose $\beta$ is a locally finite family and $x\in X$. We should find a
neighborhood $U$ of $x$ which intersects only finitely many elements from
$\alpha$. Let $y=p(x)$. Since $\beta$ is locally finite, there exists a
neighborhood $V$ of $y$ intersecting only finitely many elements
$W_{i_{1}},\ldots,W_{i_{k}}\in\beta$. Then $p^{-1}(V)$ is an open neighborhood
of $x$ intersecting only the following elements
$p^{-1}(W_{i_{1}}),\ldots,p^{-1}(W_{i_{k}})$ of $\alpha$.

Conversely, suppose $\alpha$ is locally finite and $p$ is open. Let $y\in Y$
and $x\in X$ be such that $p(x)=y$. Then there exists a neighborhood $U$ of
$x$ intersecting only finitely many elements, say
$p^{-1}(W_{i_{1}}),\ldots,p^{-1}(W_{i_{k}})$, of $\alpha$. Therefore its
saturation $S(U)=p^{-1}(p(U))$ also intersects only
$p^{-1}(W_{i_{1}}),\ldots,p^{-1}(W_{i_{k}})$.

Since $p$ is open, the image $p(U)$ is an open neighborhood of $y$. We claim
that $p(U)$ intersects only the elements $W_{i_{1}},\ldots,W_{i_{k}}$ of
$\alpha$. Indeed, if $p(U)\cap W_{i}\not=\varnothing$ for some $i\in\Lambda$,
then $p^{-1}(p(U))\cap p^{-1}(W_{i})\not=\varnothing$ which is possible only
when $i\in\\{i_{1},\ldots,i_{k}\\}$.

(5) For each $i\in\mathbb{N}$ consider the following subfamily
$\alpha_{i}=\\{\omega_{j}\\}_{j\geq i}$ of $\alpha$, so $\alpha=\alpha_{1}$
and $\alpha_{i+1}\subset\alpha_{i}$ for all $i\in\mathbb{N}$. Then each
$\alpha_{i}$ is locally finite as well, and therefore the union
$A_{i}=\mathop{\cup}\limits_{j=i}^{\infty}\omega_{i}$ is a closed subset of
$X$.

Since $X$ is normal and $\omega_{1}$ and $A_{2}$ are mutually disjoint and
closed, there exists an open neighborhood $U_{1}$ of $\omega_{1}$ such that
$\overline{U_{1}}\cap A_{2}=\varnothing$. Then $\omega_{2}$ and
$\overline{U_{1}}\cup A_{3}$ are mutually disjoint and closed, whence there
exists an open neighborhood $U_{2}$ of $\omega_{2}$ which does not intersect
$\overline{U_{1}}\cup A_{3}$. Repeating these arguments so on we will
construct for each $i\in\mathbb{N}$ an open neighborhood $U_{i}$ of
$\omega_{i}$ such that $\overline{U_{i}}$ does not intersect
$\bigl{(}\cup_{j=1}^{i-1}\overline{U_{j}}\bigr{)}\cup A_{i+1}$. Then
$\overline{U_{i}}\cap\overline{U_{j}}=\varnothing$ for all
$i\not=j\in\mathbb{N}$. ∎

######  Definition 3.2.

We will say that a partition $\Delta$ is _locally trivial_ if for each
$\omega\in\Delta$ there exists an open neighborhood $U$, a topological space
$J$, a point $t_{0}\in J$, and a homeomorphism $\phi:\omega\times J\to U$ such
that $\phi(\omega\times t)$ is an element of $\Delta$ for all $t\in J$ and
$\phi(x,t_{0})=x$ for all $x\in\omega$.

In particular, a foliation belonging to class $\mathcal{F}$ is a locally
trivial partition.

Notice that in the notation of Definition 3.2 $U$ is saturated and open in
$X$, whence its image $V=p(U)$ is open in $Y$ and we have the following
commutative diagram:

$\textstyle{\omega\times
J\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\cong}$$\scriptstyle{q_{2}}$$\textstyle{U=p^{-1}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{J\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi}$$\scriptstyle{1-1}$$\textstyle{V}$
(3.3)

where $q_{2}$ is a projection onto the second multiple and $\xi$ is the
induced one-to-one continuous map but it is not necessarily a homeomorphism.

###### Lemma 3.3

The following conditions are equivalent:

  1. (1)

the quotient map $p:X\to Y$ is a locally trivial fibration;

  2. (2)

partition $\Delta$ is locally trivial and the quotient map $p:X\to Y$ is open.

###### Proof.

(1)$\Rightarrow$(2). Suppose $p$ is a locally trivial fibration. We claim that
then $\Delta$ is locally trivial. Indeed, let $\omega\in\Delta$ and
$y=p(\omega)\in Y$. Since $p$ is locally trivial, there exists a neighborhood
$V$ of $y$ and the following commutative diagram:

$\textstyle{\omega\times
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\cong}$$\scriptstyle{q_{2}}$$\textstyle{U_{\omega}=p^{-1}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi=\mathrm{id}_{V}}$$\textstyle{V}$
(3.4)

in which $\phi$ is a homeomorphism. This diagram coincides with (3.3) for
$J=V$, and therefore $\Delta$ is a locally trivial partition.

Let us prove that $p$ is an open map. Notice that in Diagram (3.4) $q_{2}$ is
an open map as a coordinate projection. Since $\phi$ is a homeomorphism, it
follows that the restriction $p|_{U_{\omega}}$ is an open map as well. But
then $\beta=\\{U_{\omega}\\}_{\omega\in\Delta}$ is an open cover of $X$ such
that each restriction $p|_{U_{\omega}}$ is open. Therefore by (2) of Lemma 3.1
$p$ is open.

(2)$\Rightarrow$(1). Suppose $p$ is an open map and $\Delta$ is locally
trivial. We claim that then in (3.3) the map $\xi$ is open, and therefore it
is a homeomorphism. This will imply that $p$ is a locally trivial fibration.

Let $T\subset J$ be an open subset. Then $\phi\circ q_{2}^{-1}(T)$ is open in
$U$. Since $p$ is open, we get that $\xi(T)=p\circ\phi\circ q_{2}^{-1}(T)$ is
open in $V$. Thus $\xi$ is an open map. ∎

######  Definition 3.4.

An element $\omega\in\Delta$ will be called _special_ if its image
$y=p(\omega)\in Y$ is a special point of $Y$, i.e.
$y\not=\mathrm{hcl}(y):=\mathop{\cap}\limits_{V\in\beta_{y}}\overline{V}$,
where $\beta_{y}$ is the family of all neighborhoods of $y$, see Definition
2.1. Let also

$\displaystyle\mathrm{hcl}(\omega)$
$\displaystyle=\underset{N(\omega)}{\bigcap}\overline{S\left(N(\omega)\right)},$
$\displaystyle\mathrm{hcl}_{S}(\omega)$
$\displaystyle=\underset{N_{S}(\omega)}{\bigcap}\overline{N_{S}(\omega)},$

where $N(\omega)$ runs over _all open_ neighborhoods of $\omega$ and
$N_{S}(\omega)$ runs over _all saturated open_ neighborhoods of $\omega$.

###### Lemma 3.5

Let $\omega\in\Delta$ and $y=p(\omega)$. Then

$\displaystyle\mathrm{hcl}(\omega)$ $\displaystyle\ \subset\
\mathrm{hcl}_{S}(\omega)\ \subset\ p^{-1}(\mathrm{hcl}(y)).$

If $p$ is an open map, then

$\displaystyle\mathrm{hcl}(\omega)$
$\displaystyle=\mathrm{hcl}_{S}(\omega)=p^{-1}(\mathrm{hcl}(y)),$
$\displaystyle p(\mathrm{hcl}_{S}(\omega))$ $\displaystyle=\mathrm{hcl}(y).$

###### Proof.

First we establish relations between $\mathrm{hcl}(\omega)$ and
$\mathrm{hcl}_{S}(\omega)$. Notice that the family
$\mathcal{A}=\\{S(N(\omega))\\}$ of saturations of all open neighborhoods of
$\omega$ includes the family $\mathcal{B}=\\{N_{S}(\omega)\\}$ of all
saturated open neighborhoods of $\omega$. Therefore the intersection
$\mathrm{hcl}(\omega)$ of the larger family $\mathcal{A}$ is contained in the
intersection $\mathrm{hcl}_{S}(\omega)$ of the smaller family $\mathcal{B}$,
that is $\mathrm{hcl}(\omega)\ \subset\ \mathrm{hcl}_{S}(\omega)$.

If $p$ is an open map, so the saturation of an open set is open, then
$\mathcal{A}=\mathcal{B}$, and therefore
$\mathrm{hcl}(\omega)=\mathrm{hcl}_{S}(\omega)$.

Now we will describe relationships between $\mathrm{hcl}_{S}(\omega)$ and
$\mathrm{hcl}(y)$. By definition of the quotient topology on $Y$ the map $p$
induces a bijection between the families $\mathcal{B}$ and $\beta_{y}$.
Moreover, if $N_{S}(\omega)\in\mathcal{B}$ is an open saturated neighborhood
of $\omega$ and $V=p(N_{S}(\omega))$ is an open neighborhood of $y$, then, due
to continuity of $p$, we have that
$p(\overline{N_{S}(\omega)})\subset\overline{V}$. Hence
$p(\mathrm{hcl}_{S}(\omega))\subset\mathrm{hcl}(y)$, that is
$\mathrm{hcl}_{S}(\omega)\ \subset\ p^{-1}(\mathrm{hcl}(y))$.

If $p$ is open, then, due to (3.2),
$p(\overline{N_{S}(\omega)})=\overline{p(N_{S}(\omega))}=\overline{V}$, whence

$p(\mathrm{hcl}_{S}(\omega))=p\Bigl{(}\,\bigcap_{N_{S}(\omega)\in\mathcal{B}}\overline{N_{S}(\omega)}\,\Bigr{)}=p\Bigl{(}\,\bigcap_{V\in\beta_{y}}p^{-1}(\overline{V})\,\Bigr{)}=\bigcap_{V\in\beta_{y}}\overline{V}=\mathrm{hcl}(y).$
(3.5)

Finally, as $\mathrm{hcl}_{S}(\omega)$ is saturated as an intersection of
saturated sets, it follows from (3.5) that
$\mathrm{hcl}_{S}(\omega)=p^{-1}(\mathrm{hcl}_{S}(y))$. ∎

###### Lemma 3.6

Suppose that the following conditions hold true:

  1. (a)

$p:X\to Y$ is a locally trivial fibration with fiber $\mathbb{R}$;

  2. (b)

the set $\Sigma$ of special elements of $X$ is locally finite;

  3. (c)

$Y$ is a $T_{1}$-space locally homeomorphic with open subsets of $[0,1)$.

Then every connected component $Q$ of $X\setminus\Sigma$ is open in $X$ and is
foliated homeomorphic with one of the following five stripped surfaces: model
strips $\mathbb{R}\times(0,1)$, $\mathbb{R}\times[0,1)$,
$\mathbb{R}\times[0,1]$, or standard cylinder $C$, or standard Möbius band
$M$. Moreover, in the last three cases, $Q$ is also closed in $X$.

###### Proof.

By (a) and Lemma 3.3 $p$ is an open map. Therefore by Lemma 3.5
$\Sigma=p^{-1}(\mathcal{V})$ and $p(\Sigma)=\mathcal{V}$, where $\mathcal{V}$
is the set of special points of $Y$. Then by (b) and (4) of Lemma 3.1
$\mathcal{V}$ is also a locally finite family of points. Due to (c) each point
in $Y$ is closed, whence $\mathcal{V}$ is closed in $Y$.

Let $W$ be a connected component $Y\setminus\mathcal{V}$. Then $W$ is open in
$Y$ and open closed in $Y\setminus\mathcal{V}$. Therefore $Q=p^{-1}(W)$ is
open in $X$ and open closed in $X\setminus\Sigma$, i.e. $Q$ is a connected
component of $X\setminus\Sigma$. Moreover, due to (a) the restriction $p:Q\to
W$ is a locally trivial fibration with fiber $\mathbb{R}$, and by Lemma 2.3
$W$ is homeomorphic with one of the following spaces: $(0,1)$, $[0,1)$,
$[0,1]$, $S^{1}$. Therefore in the first three cases (when $W$ is
contractible) $Q$ is fiber-wise homeomorphic to a product $\mathbb{R}\times
W$, and in the last case, when $W\cong S^{1}$, $Q$ is fiber-wise homeomorphic
either with the standard cylinder $C$ or with the standard Möbius band $M$.

It remains to show that every connected component $Q$ of $X\setminus\Sigma$
can be represented as $Q=p^{-1}(W)$ for some connected component $W$ of
$Y\setminus\mathcal{V}$. Let $W=p(Q)$. We claim that $W$ is open closed in
$Y\setminus\mathcal{V}$. Indeed, let $W^{\prime}$ be the connected component
of $Y\setminus\mathcal{V}$ containing $W$. Then as noted above
$p^{-1}(W^{\prime})$ is connected and contains $Q$, whence
$Q=p^{-1}(W^{\prime})$, and so $W=W^{\prime}$. ∎

##  4\. Proof of (1) of Theorem 1.8

Let $X$ be a $2$-dimensional manifold and $\Delta$ be a $1$-dimensional
foliation on $X$ belonging to class $\mathcal{F}$ and such that the set
$\Sigma$ of special leaves of $X$ is locally finite. Let also $Y=X/\Delta$ be
the space of leaves endowed with the corresponding factor topology and $p:X\to
Y$ be the factor map.

We claim that $p$ satisfies conditions (a)–(c) of Lemma 3.6. Indeed, by Lemma
1.2 $p$ is open, and by Lemma 3.3 it is a locally trivial fibration with fiber
$\mathbb{R}$, so condition (a) holds. Condition (b) holds by assumption and
condition (c) directly follows from definition of class $\mathcal{F}$.

Therefore by Lemma 3.6 every connected component $X\setminus\Sigma$ is
foliated homeomorphic with one of the spaces: $\mathbb{R}\times(0,1)$,
$\mathbb{R}\times[0,1)$, $\mathbb{R}\times[0,1]$, $C$, $M$.

Applying the above result to the surface $X\setminus\partial X$ we get that
every connected component of $X\setminus(\Sigma\cup\partial X)$ is foliated
homeomorphic with one of the spaces: $\mathbb{R}\times(0,1)$, $C$, or $M$.
Statement (1) of Theorem 1.8 is proved.

##  5\. Trapezoids

The results of this section will be used for the proof of (2) of Theorem 1.8.

Let $c<d$ and $\alpha,\beta:(c,d]\to\mathbb{R}$ be two continuous functions
such that $\alpha(y)<\beta(y)$ for all $y\in(c,d]$. Then the subset

$T=\\{(x,y)\in\mathbb{R}^{2}\mid\alpha(y)\leq x\leq\beta(y),\ c<y\leq d\\}$

will be called a _half open trapezoid_ or simply a _trapezoid_. In this case
$[\alpha(d),\beta(d)]\times d$ is the _upper base_ of $T$, $d$ is the _level_
of the upper base, $d-c$ is the _altitude_ of $T$, and the set

$\mathrm{roof}(T):=\\{(\alpha(y),y)\\}_{y\in(c,d]}\ \cup\
[\alpha(d),\beta(d)]\times d\ \cup\ \\{(\beta(y),y)\\}_{y\in(c,d]}$

is the _roof_ of $T$, see Figure 5.1a).

|  |
---|---|---
a) Half open trapezoid |  | b) Construction of trapezoid
Figure 5.1. Half open trapezoid

Notice that if $\phi:\mathbb{R}\times(c,d]\to\mathbb{R}\times(c,d]$ is a
homeomorphism preserving second coordinate, i.e. $\phi(\mathbb{R}\times
y)=\mathbb{R}\times y$ for all $y\in(c,d]$, then $\phi(T)$ is a trapezoid as
well.

In general, $\alpha$ and $\beta$ can be non-bounded or have no limits when
$y\to c+0$. Suppose, in addition, that there exist finite or infinite limits

$\displaystyle\lim\limits_{y\to c+0}\alpha(y)$ $\displaystyle=a,$
$\displaystyle\lim\limits_{y\to c+0}\beta(y)=b$

such that $a<b$. Then $(a,b)\times c$ will be called the _(lower) base_ of
$T$. If $a$ and $b$ are finite numbers, then $T$ will be called a _trapezoid
with bounded base_ , and the set

$\overline{T}=T\cup[a,b]\times c$

will be a _closed_ trapezoid. In particular, if $\alpha$ and $\beta$ are
constant functions, then the trapezoid $T$ will be called a _rectangle_.

###### Lemma 5.1

Let $J=(a,b)\times 0\subset\mathbb{R}^{2}$ be an open interval, $N=J\ \cup\
\mathbb{R}^{2}\times(0,+\infty)$, and $U$ be an open neighborhood of $J$ in
$N$. Then there exists a half open trapezoid $T\subset U$ with base $J$, see
Figure 5.1b).

###### Proof.

Fix any two sequences
$\\{a_{i}\\}_{i=0}^{\infty},\\{b_{i}\\}_{i=0}^{\infty}\subset(a,b)$ such that
$\lim\limits_{i\to\infty}a_{i}=a$, $\lim\limits_{i\to\infty}b_{i}=b$, and

$\cdots<a_{i+1}<a_{i}<\cdots<a_{0}<b_{0}<\cdots<b_{i}<b_{i+1}<\cdots$

Let also $J_{i}=[a_{i},b_{i}]\times 0$. Since $U$ is an open neighborhood of
$J_{i}$ and $J_{i}$ is compact, there exist $r_{i}>0$ such that
$J_{i}\times[0,r_{i}]\subset U$. One can assume that
$\lim\limits_{i\to\infty}r_{i}=0$ and $\\{r_{i}\\}$ is strictly decreasing.
Now let $\alpha,\beta:(0,d]\to(0,+\infty)$ be a unique continuous function
such that for each $i\geq 0$

  * (i)

$\alpha(a_{i})=\beta(b_{i})=r_{i+1}$;

  * (ii)

the restrictions $\alpha|_{[a_{i+1},a_{i}]}$ and $\beta|_{[b_{i},b_{i+1}]}$
are linear.

Then one easily checks that the function $\alpha$ and $\beta$ are strictly
monotone and their inverses $\alpha^{-1}$ and $\beta^{-1}$ determine a half
open trapezoid $T\subset U$ with base $J$. ∎

###### Proposition 5.2

Let $T_{i}\subset\mathbb{R}\times(c,d]$, $i\in\mathbb{N}$, be a half open
trapezoid with upper base at level $d_{i}\in(c,d]$ such that $T_{i}\cap
T_{j}=\varnothing$ for $i\not=j$ and $\lim\limits_{i\to\infty}d_{i}=c$. Then
there exists a homeomorphism
$\eta:\mathbb{R}\times(c,d]\to\mathbb{R}\times(c,d]$ such that

  * (i)

$\eta(\mathbb{R}\times y)=\mathbb{R}\times y$ for all $y\in(c,d]$;

  * (ii)

$\eta(T_{i})$ is a half open rectangle.

###### Proof.

We need the following three lemmas. It will be convenient to say that for a
function $f:[a,b]\to\mathbb{R}$ its _graph_ is the subset $\\{(f(y),y)\mid
y\in[a,b]\\}\ \subset\ \mathbb{R}^{2}$, so we just swap coordinates with
respect to the usual definition. In particular, for $q\in\mathbb{R}$ a
vertical segment $q\times[a,b]$ can be regarded as a graph of a constant
function $[a,b]\mapsto q$.

###### Lemma 5.2.1

(c.f. [14, Lemma 6.1.1]). Let
$\Delta_{k}=\\{(y_{1},\ldots,y_{k})\in\mathbb{R}^{k}\mid
y_{1}<y_{2}<\ldots<y_{k}\\}$ and $q_{1}<q_{2}<\ldots<q_{k}\in\mathbb{R}$. Then
there exists a $C^{\infty}$ function
$u_{k}:\mathbb{R}\times\Delta_{k}\to\mathbb{R}$ having the following
properties:

  * (a)

the correspondence $x\mapsto u_{k}(x;y_{1},\ldots,y_{k})$ is an orientation
preserving homeomorphism $\mathbb{R}\to\mathbb{R}$ for all
$(y_{1},\ldots,y_{k})\in\Delta_{k}$;

  * (b)

$u_{k}(x;q_{1},\ldots,q_{k})=x$ for all $x\in\mathbb{R}$;

  * (c)

$u_{k}(y_{i};y_{1},\ldots,y_{k})=q_{i}$ for $i=1,\ldots,k$.

###### Proof.

The construction of $u_{k}$ is similar to [14, Lemma 6.1.1]. For instance, one
can set

$\displaystyle u_{1}(x;y_{1})$ $\displaystyle=x-y_{1}+q_{1},$ $\displaystyle
u_{2}(x;y_{1},y_{2})$
$\displaystyle=q_{1}+\frac{q_{2}-q_{1}}{y_{2}-y_{1}}(x-y_{1}).$

We leave the details for the reader. ∎

###### Lemma 5.2.2

Let $\gamma_{i}:(c,s]\to\mathbb{R}$, $i=1,\ldots,k$, be a finite family of
continuous functions such that $\gamma_{i}(y)\not=\gamma_{j}(y)$ whenever
$i\not=j$ and $y\in(c,s]$. Then there exists a homeomorphism
$\phi:\mathbb{R}\times(c,s]\to\mathbb{R}\times(c,s]$ such that

  * (1)

$\phi(\mathbb{R}\times y)=\mathbb{R}\times y$ for all $y\in(c,s]$;

  * (2)

$\phi$ is fixed on $\mathbb{R}\times s$;

  * (3)

$\phi$ maps the graph $\\{(\gamma_{i}(y),y)\mid y\in(c,s])\\}$ of
$\gamma_{i}$, $i\in\\{1,\ldots,k\\}$, onto a vertical segment
$\gamma_{i}(s)\times[a,s]$.

###### Proof.

One can assume that $\gamma_{i}<\gamma_{j}$ for $i<j$. Let
$u_{k}:\mathbb{R}\times\Delta_{k}\to\mathbb{R}$ be a function from Lemma 5.2.1
constructed for the numbers $q_{i}=\gamma_{i}(s)$, $i\in\\{1,\ldots,k\\}$.
Then a homeomorphism $\phi$ satisfying (1)-(3) can be defined by the following
formula:

$\phi(x,y)=\bigl{(}u_{k}(x;\gamma_{1}(y),\ldots,\gamma_{k}(y)),\ y\bigr{)}.$

Indeed, (1) is evident. Moreover, due to property (b) of $u_{k}$ we have that

$\phi(x,s)=\bigl{(}u_{k}(x;\gamma_{1}(s),\ldots,\gamma_{k}(s)),\
s\bigr{)}=(x,s)$

which proves (2). Finally, by property (c) of $u_{k}$

$\phi(\gamma_{i}(y),y)=\bigl{(}u_{k}(\gamma_{i}(y);\gamma_{1}(y),\ldots,\gamma_{k}(y)),\
y\bigr{)}=(\gamma_{i}(s),y),$

so (3) is also satisfied. ∎

###### Lemma 5.2.3

Let $\\{d_{i}\\}_{i\in\mathbb{N}}\subset(c,d]$ be a sequence with
$\lim\limits_{i\to\infty}d_{i}=c$, and for each $i\in\mathbb{N}$ let
$\gamma_{i}:(c,d_{i}]\to\mathbb{R}$ be a continuous function such that the
graphs of $\gamma_{i}$ and $\gamma_{j}$ are mutually disjoint for $i\not=j$.
Then there exists a homeomorphism
$\eta:\mathbb{R}\times(c,d]\to\mathbb{R}\times(c,d]$ such that

  * (i)

$\eta(\mathbb{R}\times y)=\mathbb{R}\times y$ for all $y\in(c,d]$;

  * (ii)

$\eta$ maps the graph $\\{(\gamma_{i}(y),y)\mid y\in(c,d_{i}]\\}$ of
$\gamma_{i}$ onto a vertical segment $q_{i}\times(c,d_{i}]$ for some
$q_{i}\in\mathbb{R}$, $i\in\mathbb{N}$.

###### Proof.

One can assume, in addition, that $\\{d_{i}\\}_{i\in\mathbb{N}}$ is non-
increasing. Let us remove repeating elements from
$\\{d_{i}\\}_{i\in\mathbb{N}}$ and denote the obtained sequence by
$\\{s_{i}\\}_{i\in\mathbb{N}}$. Thus there is an increasing sequence of
indices $1=j_{1}<j_{2}<\cdots<j_{n}<\cdots$ such that

$s_{i}=d_{j_{i}}=d_{j_{i}+1}=\cdots=d_{j_{i+1}-1}\ >\
s_{i+1}=d_{j_{i+1}}=\cdots.$

Then by Lemma 5.2.2 there exists a homeomorphism
$\phi_{1}:\mathbb{R}\times(c,s_{1}]\to\mathbb{R}\times(c,s_{1}]$ preserving
second coordinate and sending the graphs of functions
$\gamma_{j_{1}},\ldots,\gamma_{j_{2}-1}$ onto vertical segments. Let us extend
$\phi_{1}$ by the identity on $\mathbb{R}\times[s_{1},d]$ to a homeomorphism
of all of $\mathbb{R}\times(c,d]$.

Denote by $\gamma_{i}^{1}$ the image of the graph of $\gamma_{i}$ under
$\phi_{1}$. Then again there exists a homeomorphism
$\phi_{2}:\mathbb{R}\times(c,d]\to\mathbb{R}\times(c,d]$ preserving second
coordinate, fixed on $\mathbb{R}\times[s_{2},d]$, and sending the graphs of
functions $\gamma^{1}_{j_{1}},\ldots,\gamma^{1}_{j_{3}-1}$ onto vertical
segments.

Hence the composition $\phi_{2}\circ\phi_{1}$ preserves second coordinate and
sends the graphs of functions $\gamma_{j_{1}},\ldots,\gamma_{j_{3}-1}$ onto
vertical segments. Denote by $\gamma^{2}_{i}$ the graph of the function
$\gamma_{i}$ under $\phi_{2}\circ\phi_{1}$.

Then by similar arguments, we will construct an infinite family of
homeomorphisms $\phi_{1},\ldots,\phi_{k},\ldots$ of $\mathbb{R}\times(c,d]$
such that each $\phi_{k}$ preserves second coordinate, is fixed on
$\mathbb{R}\times[s_{k},d]$, and sends the graphs of functions
$\gamma^{1}_{j_{1}},\ldots,\gamma^{1}_{j_{k+1}-1}$ onto vertical segments.

Since $\lim\limits_{i\to\infty}s_{i}=c$, it follows that the infinite
composition

$\eta=\cdots\circ\phi_{m}\circ\phi_{m-1}\circ\cdots\circ\phi_{1}:\mathbb{R}\times(c,d]\to\mathbb{R}\times(c,d]$

is a well defined homeomorphism satisfying the statement of lemma. ∎

To deduce Proposition 5.2 assume that $T_{i}$ is defined by functions
$\alpha_{i},\beta_{i}:(c,d_{i}]\to\mathbb{R}$. Denote
$\gamma_{2i-1}=\alpha_{i}$ and $\gamma_{2i}=\beta_{i}$. Then existence of
$\eta$ is guaranteed by Lemma 5.2.3. ∎

### Level-preserving homeomorphisms between trapezoids.

Let $q_{2}:\mathbb{R}^{2}\to\mathbb{R}$, $q_{2}(x,y)=y$, be the standard
projection onto the second coordinate and

$\displaystyle S$ $\displaystyle=\\{(x,y)\in\mathbb{R}\mid\alpha(y)\leq
x\leq\beta(y),\ a<y\leq b\\},$ $\displaystyle T$
$\displaystyle=\\{(x,y)\in\mathbb{R}\mid\gamma(y)\leq x\leq\delta(y),\ c<y\leq
d\\}$

be two trapezoids with finite bases, where $\alpha,\beta:(a,b]\to\mathbb{R}$
and $\gamma,\delta:(c,d]\to\mathbb{R}$ are continuous functions such that
$\alpha<\beta$ and $\gamma<\delta$.

Let $A\subset S$ and $B\subset T$ be two subsets. Then a map $\xi:A\to B$ will
be called _level-preserving_ whenever

$q_{2}\circ\xi(x,y)=q_{2}\circ\xi(x^{\prime},y)$

for all $x,x^{\prime},y$ such that $(x,y),(x^{\prime},y)\in A$.

###### Lemma 5.3

Every level-preserving homeomorphism $\xi:\mathrm{roof}(S)\to\mathrm{roof}(T)$
between roofs of trapezoids extends to a level-preserving homeomorphism
$\xi:S\to T$. Moreover, if $S$ and $T$ have finite bases, then $\xi$ also
extends to a level-preserving homeomorphism $\xi:\overline{S}\to\overline{T}$
between their closures.

###### Proof.

As $\xi$ is level-preserving, we have a well defined homeomorphism
$\sigma:(a,b]\to(c,d]$ given by $\sigma(y)=q_{2}\circ\xi(\alpha(y),y)$. Then
$\xi$ extends to a homeomorphism $S\to T$ by

$\xi(x,y)=\left(\gamma(\sigma(y))+\frac{\delta(\sigma(y))-\gamma(\sigma(y))}{\beta(y)-\alpha(y)}(x-\alpha(y)),\
\sigma(y)\right).$

Moreover, if in addition $S$ and $T$ have finite bases, so $\alpha$ and
$\beta$ are defined and continuous on $[a,b]$ and $\gamma$ and $\delta$ are
defined and continuous on $[c,d]$, then the same formulas define
homeomorphisms $\sigma:[a,b]\to[c,d]$ and $\xi:\overline{S}\to\overline{T}$. ∎

##  6\. Proof of (2) of Theorem 1.8

Let $\Delta$ be a partition on $X$ of class $\mathcal{F}$ such that the family
$\Sigma$ of all special leaves is locally finite. Let also
$\bar{\Sigma}=\Sigma\cup\partial X$ be the union of all special and boundary
leaves of $X$, $Q$ be a connected component of $X\setminus\bar{\Sigma}$
homeomorphic with an open model strip, and $\phi:\mathbb{R}\times(-1,1)\to Q$
be a foliated homeomorphism. Denote, see Figure 6.1:

$\displaystyle A$ $\displaystyle=\phi\bigl{(}\mathbb{R}\times(-1,0]\bigr{)},$
$\displaystyle K$ $\displaystyle=\phi\bigl{(}\mathbb{R}\times 0\bigr{)},$
$\displaystyle B$ $\displaystyle=\phi\bigl{(}\mathbb{R}\times[0,1)\bigr{)}.$
(6.1)

We should prove that the closures $\overline{A}$ and $\overline{B}$ are
foliated homeomorphic to some model strips. It suffices to prove this only for
$\overline{A}$.

Figure 6.1.

###### Lemma 6.1

1) $\overline{A}\setminus Q=\overline{A}\setminus A$ and
$\overline{B}\setminus Q=\overline{B}\setminus B$.

2) $\overline{Q}\setminus Q=(\overline{A}\setminus
A)\cup(\overline{B}\setminus B)\ \subset\ \bar{\Sigma}$.

3) Let $\omega$ be a leaf in $\overline{A}\setminus A$, $J=(-1,1)\times
0\subset\mathbb{R}^{2}$, and $N=J\ \cup\ \mathbb{R}\times(0,1]$.

  1. (a)

Then $Q\cup\omega$ is open in $\overline{Q}$ and $A\cup\omega$ is open in
$\overline{A}$.

  2. (b)

There exists a foliated homeomorphism $\psi:N\to A\cup\omega$ such that
$\psi(J)=\omega$.

  3. (c)

Let $U\subset X$ be an open neighborhood of $\omega$ and
$T\subset\psi^{-1}(U)$ be a subset with compact closure such that

$\displaystyle\overline{U}\ \cap\ (\overline{A}\setminus A)\ =\ \omega,$
$\displaystyle J\ \subset\ (\overline{T}\setminus T)\ \subset\
\mathbb{R}\times 0.$

Then $\psi(T)$ is closed in $X$. In particular, if $T$ is a trapezoid with
base $J$, then $\psi(T\cup J)$ is closed in $X$.

###### Proof.

1) Denote $A^{o}=A\setminus K$ and $B^{o}=B\setminus K$. Then
$K\subset\overline{A^{o}}$, so

$\overline{A}=\overline{A^{o}\cup K}=\overline{A^{o}}\cup K=\overline{A^{o}}.$

Moreover, as $A^{o}$ and $B^{o}$ are open in $X$ and disjoint, we get that
$\overline{A}\cap B^{o}=\overline{A^{o}}\cap B^{o}=\varnothing$, whence
$\overline{A}\setminus Q=\overline{A}\setminus(A\cup
B^{o})=\overline{A}\setminus A$. The proof for $B$ is similar.

2) It follows from (1) that $\overline{Q}\setminus Q=(\overline{A}\setminus
Q)\cup(\overline{B}\setminus Q)=(\overline{A}\setminus
A)\cup(\overline{B}\setminus B)$.

Let us prove that $\overline{Q}\setminus Q\subset\bar{\Sigma}$. Suppose
$\overline{Q}\setminus Q\not\subset\bar{\Sigma}$. Then there exists a
connected component $P$ of $X\setminus\bar{\Sigma}$ distinct from $Q$ and such
that $\overline{Q}\cap P\not=\varnothing$. But $P$ is open in $X$, whence
$P\cap Q\not=\varnothing$ and so $P=Q$ which contradicts to the assumption.

3a) Notice that the family $\bar{\Sigma}\setminus\\{\omega\\}$ is locally
finite as well as $\bar{\Sigma}$. Therefore the set

$W:=X\setminus(\bar{\Sigma}\setminus\omega)=(X\setminus\bar{\Sigma})\cup\omega$

is open in $X$. Due to 2), $Q=\overline{Q}\cap(X\setminus\bar{\Sigma})$,
whence
$Q\cup\omega=\overline{Q}\cap\bigl{(}(X\setminus\bar{\Sigma})\cup\omega\bigr{)}=\overline{Q}\cap
W$ is open in $\overline{Q}$.

Similarly, due to 1), $A=\overline{A}\cap Q$, whence
$A\cup\omega=\overline{A}\cap\overline{Q}\cap\bigl{(}(X\setminus\bar{\Sigma})\cup\omega\bigr{)}=\overline{A}\cap
W$ is open in $\overline{A}$.

3b) Notice that $A\cup\omega$ is saturated and by Lemma 2.4 $p(A\cup\omega)$
is homeomorphic with $[0,1]$. Since $p:A\cup\omega\to p(A\cup\omega)$ is a
locally trivial fibration with fiber $\mathbb{R}$, we obtain that
$A\cup\omega$ is foliated homeomorphic with $\mathbb{R}\times[0,1]$ and
therefore with $N$.

3c) It suffices to prove that $\psi(T)$ is closed in $\overline{A}\setminus U$
being a closed subset of $X$, which will imply that $\psi(T)$ is closed in $X$
as well.

Let $\\{z_{i}\\}_{i\in\mathbb{N}}\subset\psi(T)$ be a sequence converging to
some $z\in\overline{A}$. We should prove that $z\in\psi(T)$ as well. Let
$(x_{i},y_{i})=\psi^{-1}(z_{i})\in T$. Since $\overline{T}$ is compact, one
can assume that $\\{(x_{i},y_{i})\\}$ converges to some
$(\bar{x},\bar{y})\in\overline{T}$.

If $(\bar{x},\bar{y})\in T$, then
$z=\lim\limits_{i\to\infty}z_{i}=\lim\limits_{i\to\infty}\psi(x_{i},y_{i})=\psi(\bar{x},\bar{y})\in\psi(T)$.
Otherwise, we have that $(\bar{x},\bar{y})\in\overline{T}\setminus
T\subset\mathbb{R}\times 0$, so $\bar{y}=0$, and thus
$\lim\limits_{i\to\infty}y_{i}=\bar{y}=0$. This implies that $z\not\in
A=\psi\bigl{(}\mathbb{R}\times(0,1]\bigr{)}$. Hence
$z\in\overline{U}\cap(\overline{A}\setminus A)=\omega=\psi(J)\subset\psi(T)$.
∎

Due to (5) of Lemma 3.1 there exists a family
$\mathcal{U}=\\{U_{\omega}\\}_{\omega\in\bar{\Sigma}}$ of neighborhoods of
elements of $\bar{\Sigma}$ such that the closures of elements of $\mathcal{U}$
are pairwise disjoint in $X$.

Let $\\{\omega_{i}\\}_{i\in\Lambda}$ be all the leaves contained in
$\overline{A}\setminus A$. Then $\Lambda$ is at most countable set and one can
assume that either $\Lambda=\\{1,\ldots,k\\}$ for some finite $k$ or
$\Lambda=\mathbb{N}$.

By Lemma 6.1 for each $i\in\Lambda$ there exists a foliated homeomorphism
$\phi_{i}:N\to A\cup\omega_{i}$ such that $\psi_{i}(J)=\omega_{i}$.

Then $\psi_{i}^{-1}(U_{\omega_{i}})$ is an open neighborhood of
$J=(-1,1)\times 0$, whence by Lemma 5.1 there exists a trapezoid
$T_{i}\,\subset\,\psi_{i}^{-1}(U_{\omega_{i}})\,\cap\,\mathbb{R}\times(0,1)$
with base $J$. Put

$\widehat{T}_{i}=T_{i}\cup J.$

Then by Lemma 6.1 $\psi_{i}(\widehat{T}_{i})$ is closed in $\overline{A}$.

Figure 6.2.

Denote $S_{i}=\phi^{-1}\circ\psi_{i}(T_{i})$. Then $\\{S_{i}\mid
i\in\Lambda\\}$ is a family of trapezoids in $\mathbb{R}\times(-1,0]$. Assume
that the upper base of $S_{i}$ in contained in $\mathbb{R}\times d_{i}$ for
some $d_{i}\in(-1,0)$. If $\Lambda$ is infinite, then decreasing, if
necessary, $T_{i}$ (and therefore $S_{i}$), one can assume that
$\lim\limits_{i\to\infty}d_{i}=-1$. Then by Proposition 5.2 one can change
$\phi$ so that $S_{i}=[a_{i},b_{i}]\times(-1,d_{i}]$ is a “half open
rectangle” for some $a_{i},b_{i}\in\mathbb{R}$. Then
$[a_{i},b_{i}]\cap[a_{j},b_{j}]=\varnothing$ for $i\not=j\in\Lambda$. Let also
$J_{i}=(a_{i},b_{i})\times\\{-1\\}$ and

$M\ :=\ \mathbb{R}\times(-1,0]\ \bigcup\
\mathop{\cup}\limits_{i\in\Lambda}J_{i}.$

Then $M$ is a _half model strip_. Our aim is to construct a foliated
homeomorphism between $M$ and $\overline{A}$.

Denote $\widehat{S}_{i}=S_{i}\cup J_{i}$, $i\in\Lambda$, and

$Z\,:=\,M\,\setminus\,\mathop{\cup}\limits_{i\in\Lambda}\bigl{(}\widehat{S}_{i}\setminus\mathrm{roof}(S_{i})\bigr{)}\
\subset\ \mathbb{R}\times(0,1].$

###### Lemma 6.2

$\\{Z\\}\cup\\{\widehat{S}_{i}\\}_{i\in\Lambda}$ is a locally finite cover of
$M$ by closed sets.

###### Proof.

It is evident, that $\widehat{S}_{i}$ is closed in $M$. Moreover
$\widehat{S}_{i}\setminus\mathrm{roof}(S_{i})$ is open in $M$, whence $Z$ is
closed in $M$ as well. Therefore it remains only to show that each $z=(x,y)\in
M$ has an open neighborhood $V$ intersecting only finitely many elements
$\widehat{S}_{i}$.

If $y=-1$, then $z\in(a_{i},b_{i})\times\\{-1\\}\subset\widehat{S}_{i}$ for
some $i\in\Lambda$. Hence $V=\widehat{S}_{i}\setminus\mathrm{roof}(S_{i})$ is
an open neighborhood of $z$ in $N$ intersecting only $\widehat{S}_{i}$.

Suppose that $y>-1$. Fix any $t$ such that $-1<t<y$. Then
$V=\mathbb{R}\times(t,0]$ is an open neighborhood of $z$ in $M$. By assumption
$\lim\limits_{i\to\infty}d_{i}=-1$, whence there exists $n>0$ such that
$-1<d_{i}<t$ for all $i>n$, and so $\widehat{S}_{i}\cap V=\varnothing$. ∎

###### Lemma 6.3

$\\{\phi(Z)\\}\cup\\{\psi_{i}(\widehat{T}_{i})\\}_{i\in\Lambda}$ is a locally
finite cover of $\overline{A}$ by closed sets.

###### Proof.

By 3c) of Lemma 6.1 each $\psi_{i}(\widehat{T}_{i})$ is closed in $X$.
Furthermore,

$\displaystyle\phi(Z)$
$\displaystyle=\phi\Bigl{(}M\,\setminus\,\mathop{\cup}\limits_{i\in\Lambda}\,\bigl{(}\widehat{S}_{i}\setminus\mathrm{roof}(S_{i})\bigr{)}\Bigr{)}=\overline{A}\,\setminus\,\mathop{\cup}\limits_{i\in\Lambda}\,\psi_{i}\bigl{(}\widehat{T}_{i}\setminus\mathrm{roof}(T_{i})\bigr{)},$

and it is also evident that $\widehat{T}_{i}\setminus\mathrm{roof}(T_{i})$ is
open in $N$. But due to 3b) of Lemma 6.1 $\psi_{i}$ is a homeomorphism of $N$
onto the open subset $A\cup\omega_{i}$ of $\overline{A}$. Therefore
$\psi_{i}(\widehat{T}_{i}\setminus\mathrm{roof}(T_{i}))$ is open in
$\overline{A}$, whence $\phi(Z)$ is closed in $\overline{A}$.

It remains to show that $\\{\psi_{i}(\widehat{T}_{i})\\}_{i\in\Lambda}$ is a
locally finite family. Let $q\in\overline{A}$.

If $q\in\omega_{i}$, then $U_{\omega_{i}}$ is an open neighborhood of $q$
intersecting only one set $\psi_{i}(\widehat{T}_{i})$.

Suppose $q\in\overline{A}\setminus A$ and let
$z=(x,y)=\phi^{-1}(q)\in\mathbb{R}\times(-1,0]\subset M$. Then by Lemma 6.2
there exists an open neighborhood $V$ of $z$ in $\mathbb{R}\times(-1,0]$
intersecting only finitely many $\widehat{S}_{i}$. But the map
$\phi:\mathbb{R}\times(-1,0]\to A$ is a homeomorphism, whence $\phi(V)$ is an
open neighborhood of $q$ in $A$ intersecting only finitely many
$\psi_{i}(\widehat{T}_{i})=\phi(S_{i})\cup\omega_{i}$. ∎

Notice that the composition $\psi^{-1}\circ\phi|_{S_{i}}:S_{i}\to T_{i}$ is a
level-preserving homeomorphism, however in general it can not be extended to a
homeomorphism between their bases. Nevertheless, $\psi^{-1}\circ\phi$ yields a
level-preserving homeomorphism $\mathrm{roof}(S_{i})\to\mathrm{roof}(T_{i})$,
and therefore by Lemma 5.3 it extends to a level-preserving homeomorphism
$\xi_{i}:\overline{S_{i}}\to\overline{T_{i}}$.

Now define the following map $\eta:M\to\overline{A}$ by

$\eta(z)=\begin{cases}\psi_{i}\circ\xi_{i}(z),&z\in\widehat{S}_{i}\ \text{for
some }i\in\Lambda,\\\ \phi(z),&z\in Z.\end{cases}$

We claim that $\eta$ is the required homeomorphism.

Indeed, evidently, $\eta$ is a bijection. Furthermore, by Lemma 6.2
$\\{Z\\}\cup\\{\widehat{S}_{i}\\}_{i\in\Lambda}$ is a locally finite closed
cover of $M$, and by Lemma 6.3 their images
$\\{\phi(Z)\\}\cup\\{\psi_{i}(\widehat{T}_{i})\\}_{i\in\Lambda}$ constitute a
locally finite closed cover of $\overline{A}$. Finally, the restrictions
$\eta|_{Z}$ and $\eta|_{\widehat{S}_{i}}$ are homeomorphisms. Then by (6) of
Lemma 3.1 $\eta$ is a homeomorphism. Theorem 1.8 is completed.

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