One-dimensional foliations on topological manifolds

Let $X$ be an $(n+1)$-dimensional manifold, $\Delta$ be a one-dimensional foliation on $X$, and $p: X \to X / \Delta$ be a quotient map. We will say that a leaf $\omega$ of $\Delta$ is special whenever the space of leaves $X / \Delta$ is not Hausdorff at $\omega$. We present necessary and sufficient conditions for the map $p: X \to X / \Delta$ to be a locally trivial fibration under assumptions that all leaves of $\Delta$ are non-compact and the family of all special leaves of $\Delta$ is locally finite.


Introduction
Study of the topological structure of flow lines foliations has a long history and leads back to H. Poincarè. The question when a partition into curves is a foliation was considered by H. Whitney [46], [47]. In two-dimensional case one-dimensional foliations appeared as level-sets of pseudo-harmonic functions in W. Kaplan [20], [21].
Let ∆ be a one-dimensional foliation on R 2 , and R 2 /∆ be the space of leaves endowed with the quotient topology. Notice that R 2 /∆ is usually non-Hausdorff. W. Kaplan [20] showed that (1) the quotient map p : R 2 → R 2 /∆ is a locally trivial fibration with fiber R; (2) there exists at most countably many leaves {ω i } i∈A of ∆ such that the complement R 2 \ {ω i } i∈A is a disjoint union j∈B S j , where each S j is homeomorphic with (0, 1) × R so that the lines t × R, t ∈ (0, 1), correspond to the leaves of ∆; (3) there exists a pseudoharmonic function (without singularities) f : R 2 → R whose foliation by connected components of level-sets coincides with ∆. See also W. Boothby [5], [6], M. Morse and J. Jenkins [16], [17], [18], [19] and M. Morse [33], [32] for extensions of Kaplan's results to foliations with singularities.
Later C. Godbillon and G. Reeb [9] classified locally trivial fibrations over a non-Hausdorff letter Y . Though they considered a very special case their methods clarify the general situation.
The question when for an arbitrary k-dimensional foliation ∆ on X the quotient map p : X → X/∆ has homotopy lifting properties was considered in C. Godbillon [10], see also G. Meigniez [29] and [30] for the criterion when p is a Serre fibration or a locally trivial fibration but mostly in smooth category. J. Harrison [14] studied similar problem concerning geodesic flows without compact orbits. Also foliations by flow lines on 3-manifolds are classified by S. Matsumoto [27].
In recent years a progress in the theory of Hamiltonial dynamical systems of small degrees of freedom increased an interest to the structure of level-sets functions on surfaces, see e.g.
In [26] the authors extended Kaplan's result (2) to foliations on arbitrary non-compact surfaces X. Namely, under certain assumptions including (1), i.e. that p : X → X/∆ is a locally trivial fibration, the topological structure of the closures S j of strips S j was described.
In the present paper we consider an arbitrary one-dimensional foliation ∆ with all noncompact leaves on a topological manifold X. Our main result gives necessary and sufficient conditions for the quotient map p : X → X/∆ to be a locally trivial fibration, see Theorem 2.8.
As mentioned above such types of questions were extensively studied. However, the essentially new features of Theorem 2.8 in comparison e.g. with [10], [30] and others, is that we work in C 0 category only and give a characterization in terms of the topology of the quotient space X/∆.

One-dimensional foliations
is called a plaque of this foliated chart. Definition 2.2 (cf. [7,45]). Let ∆ = {ω α | α ∈ A} be a partition of X into pathconnected subsets ω α of X. Suppose that X admits an atlas {U i , ϕ i } i∈Λ of foliated charts of codimension n such that, for each α ∈ A and each i ∈ Λ, every pathcomponent of a set ω α ∩ U i is a plaque. Then ∆ is said to be a foliation of X of dimension 1 (and codimension n) and {U i , ϕ i } i∈Λ is called a foliated atlas associated to ∆. Each ω α is called a leaf of the foliation and the pair (X, ∆) is called a foliated manifold. Remark 2.3. In [20] one-dimensional foliations on the plane were also called regular families of curves.
In what follows we will assume that X is endowed with some 1-dimensional foliation ∆. We will also consider only foliated charts included into some (maximal) foliated atlas associated to ∆.
Recall that a continuous map f : A → B is called proper whenever for each compact K ⊂ B its inverse image f −1 (K) is compact. The following lemma is easy and we leave it for the reader. (m) there exists an embedding φ : R → X with φ(R) = ω; (p) there exists a proper injective continuous map φ : R → X with φ(R) = ω; (p) any injective continuous map φ : R → X with φ(R) = ω is proper; (c) ω is a closed subset of X.
Then the following equivalences hold true: A leaf ω satisfying condition (p) of Lemma 2.4 will be said to be properly embedded 1 . The union of all leaves of ∆ intersecting a subset U ⊂ X is called the saturation of U and denoted by S(U ). The following lemma is easy to prove. Lemma 2.5. [11,Proposition 1.5], [45,Theorem 4.10 Let ∆ i be a 1-foliation on X i , i = 1, 2. Then an embedding ψ : X 1 → X 2 will be called foliated whenever ψ(ω) is contained in some leaf of ∆ 2 for each leaf ω ∈ ∆ 1 .
In particular, if ϕ : U → (a, b) × B n is a foliated chart as in Definition 2.1, then its inverse ψ = ϕ −1 : (a, b) × B n → X is an open foliated embedding. In this case the set Space of leaves. Let Y = X/∆ be the space of leaves and p : X → Y be the corresponding quotient map. Endow Y with the quotient topology with respect to p.
Notice that for a subset U ⊂ X its saturation is S(U ) = p −1 p(U ) . In particular, Lemma 2.5 means that p is an open map.
Evidently, Y is a T 1 -space if and only if each leaf of ∆ is a closed subset of X. However, in general, Y is not a Hausdorff space.
Special points. Let u ∈ Y be a point and β u be a base of neighborhoods of u. Then the following set hcl(u) := ∩ V ∈βu V will be called the Hausdorff closure of u. A point u will be called special 2 if u = hcl(u). Notice that u ∈ hcl(v) if and only if any two neighborhoods of u and v intersect. The latter statement is symmetric with respect to u and v, and so it is equivalent to the assumption v ∈ hcl(u). However, one easily checks that the property "belong to Hausdorff closure" is not transitive.
Evidently, Y is Hausdorff if and only if u = hcl(u) for all u ∈ Y , that is when Y has no special points. The set of all special points of Y will be denoted by V.
We will say that a leaf ω of ∆ is special if p(ω) is a special point of Y . In particular, Σ := p −1 (V) is the set of all special leaves of ∆.
The following lemma gives a characterization of special leaves and extends [9, Proposition 4].
Lemma 2.6. Let ω ∈ ∆ be a leaf and u = p(ω) be the corresponding point in Y . Then the following conditions equivalent: (1) u is a special point of Y , and so ω is a special leaf of ∆; (2) there exists a point v ∈ hcl(u) distinct from u and a sequence {w i } i∈N converging to both points u and v; (3) there exist two sequences {x i } i∈N and {y i } i∈N in X such that x i and y i belong to the same leaf for all i ∈ N, that is p(x i ) = p(y i ), and Proof. Equivalence (1)⇔(2) is well known and easy.
(3)⇒(2). Denote w i = p(x i ) = p(y i ), i ∈ N and v = p(y). Then, by continuity of p, the sequence {w i } i∈N converges to distinct points u and v. In particular, v ∈ hcl(u).
(2)⇒(3). Choose any points x ∈ ω and y ∈ ω := p −1 (v), and let {U i } i∈N and {V i } i∈N be countable bases of topology on X at x and y respectively. Since p is open, p(U i ) and p(V i ) are open neighborhoods of u and v respectively. But these points are special and u ∈ hcl(v), Hence there exist x i ∈ U i and y i ∈ V i such that p(x i ) = p(y i ). Then {x i } i∈N and {y i } i∈N converge to x and y respectively.
images γ(u) and γ(v) of these points belong to distinct leaves of ∆. If x ∈ γ(V ) and ω is a leaf of ∆ containing x, then we will also say that γ passes through x as well as through ω.
The aim of this paper is to present necessary and sufficient conditions for the map p to be a locally trivial fibration under assumption that all leaves of ∆ are non-compact.
Theorem 2.8. Let X be an (n + 1)-dimensional manifold and ∆ be a one-dimensional foliation on X. Suppose that all leaves of ∆ are non-compact and the family of all special leaves of ∆ is locally finite. Then the following conditions are equivalent.
(1) The quotient map p : X → X/∆ is a locally trivial fibration with fiber R and Y is locally homeomorphic with R n + (though it is not necessary a Hausdorff space). Remark 2.9. It is proved in [10, Chapter III, Propostion 4 and Corollary] that for arbitrary p-dimensional foliation ∆ then the quotient map p : X → X/∆ is a Serre fibration whenever it satisfies a certain variant of homotopy extension property and either has a local section at each point or the quotient X/∆ is a (possibly non-Hausdorff manifold). See also [29] and [30] for extensions.
Our Theorem 2.8 claims that for one-dimensional foliations ∆ with locally finite family of special leaves existence of cross sections with open subsets of R n + implies that p is even a locally trivial fibration and X/∆ is a possibly non-Hausdorff manifold.
Remark 2.10. Equivalence between (1) and (2) for dim X = 2 is proved in [26] without assumption that Y is locally homeomorphic with R. Also in [13, §2.2, Proposition 1] it is show that X/∆ is a 1-manifold for one-dimensional foliation on R 2 .
Remark 2.11. R. H. Bing [2], [3] constructed a non-manifold B ⊂ R 4 such that R × B is homeomorphic with R 4 . In other words, R 4 admits a trivial partition into open arcs (being not a foliation) such that the quotient space B is not a 3-manifold. That example was improved by many authors, see e.g R. Rosen [38], J. Kim [22], J. Bailey [1], L. Rubin [39]. Remark 2.12. E. Dyer and M. Hamstrom [8] studied so called completely regular mappings p : X → Y between metric spaces such that the inverse images of all points are in a certain sense "uniformly homeomorphic", and get sufficient conditions when such a map is equivalent to a trivial fibration, see [8,Theorem 7], and also [28], [40] for generalizations. We consider here a similar problem, but now the space Y is not even Hausdorff, and we gave conditions when p is a locally trivial fibration.
The following statement is proved in [43,Theorem 1] for continuous functions f : R 2 → R, and in [30, item 3 at the end of page 3778] for smooth case. Proof. We claim that ∆ contains no special leaves and each leaf admits a cross section. Then it will follow from Theorem 2.8 that f is a locally trivial fibration with fiber R.
Absence of special leaves. Let Y = M/∆ be the space of leaves endowed with the corresponding factor topology. Then f can be written as a composition of the following maps where θ is the induced continuous bijection. Since N is Hausdorff, it follows that so is Y , and therefore Y contains no special points. Hence ∆ contains no special leaves.
In fact, Theorem 2.8 is an easy consequence of the following statements: Lemma 4.6. Let ω 0 be a leaf of ∆. Suppose that for each leaf ω of ∆ contained in S(ω 0 ) there exists a cross section γ passing through ω. Then ω 0 is properly embedded.
Theorem 2.14. Let γ : V → X be a cross section intersecting only leaves being simultaneously non-compact, properly embedded, and non-special. Then the saturation S(γ(V )) is open and foliated homeomorphic with R × V .
The proof of Theorem 2.8 will be given in §3. In §4 we will prove some general preliminary results concerning one-dimensional C 0 foliations being well known for smooth case. In particular we will prove Lemma 4.6. §5 is devoted to the proof of Theorem 2.14 using E. Michael's theorems about selections of multivalued maps.

Proof of Theorem 2.8
(1)⇒(2), (3). Suppose the quotient map p : X → Y is a locally trivial fibration with fiber R and Y is locally homeomorphic with R n + . This means that for each ω ∈ ∆ there exist • an open neighborhood V ⊂ Y of its image u = p(ω) homeomorphic with an open subset of R n + and • a foliated homeomorphism ψ : R × V → p −1 (V ).
is an open and saturated neighborhood of ω and ψ is a foliated homeomorphism required by (2).
(3)⇒(2). Suppose each leaf of ∆ admits a local cross section. Then it follows from Lemma 4.6 that all leaves of ∆ are properly embedded. Let Σ be a family of all special leaves and σ ∈ Σ be a special leaf.
Since each leaf is closed and Σ is a locally finite family, it follows that Σ \ σ is a closed set, whence X = (X \ Σ) ∪ σ is open and saturated and contains no special leaves. Moreover, since each leaf in X admits a local cross section, it follows from Theorem 2.14 that each leaf continuous and open bijection, and so it is a homeomorphism. Thus Y is locally homeomorphic with R n + and the map p • ψ : R × V → U u is a trivialization of p over U u , so p is a locally trivial fibration with fiber R. Theorem 2.8 is completed.

Preliminaries
In this section we will assume that X is an (n + 1)-dimensional manifold with ∂X = ∅ and ∆ is a one-dimensional foliation on X.
Some statements in this section are well known for C 1 foliations e.g. [30], and some of them are proved for C 0 case but for the foliations on R 2 , see e.g. W. Kaplan [20]. However we did not find good exposition in the literature for general C 0 foliations needed in our case and therefore short proofs will be presented. This will also make the paper self-contained.
It will be convenient to regard the graph of a function f : X → R as the following subset of R × X. Thus we switch the coordinates.
be the graph of f i . Let also c 1 < c 2 < · · · < c k ∈ (a, b) be any increasing k-tuple of numbers. We leave the details for the reader, see Proof. Let π : (a, b) × R n → R n be the standard projection. Then the assumption that γ is a cross section means that the composition is an injective map between open subsets of R n . Hence by Brouwer's theorem on domain invariance, e.g. [15], π • γ(W ) is an open neighborhood of 0 in R n . Therefore we get an open embedding satisfying (i) and (iii). Then ψ −1 (γ(W )) ⊂ (a, b) × W can be regarded as a graph of certain continuous function W → (a, b). Hence we get from Lemma 4.1 that ψ can be composed with a foliated homeomorphism of (a, b) × W to satisfy (ii), that is to make ψ −1 (γ(W )) being the graph of the constant function W → c. Moreover, statements (b) and (d) of Lemma 4.1 allow to preserve properties (i) and (iii) respectively.

Lemma 4.3.
Let ω be a leaf of ∆, J 1 , J 2 ⊂ ω be two compact segments such that J 1 ∩ J 2 is a point, V be an open n-disk, ε > 0, and be two open foliated embeddings such that Proof. Notice that the assumption that the union of the images of ψ 1 and ψ 2 does not contain compact leaves of ∆ implies that for any u, v ∈ V the union of the arcs does not contain a non-trivial loop, so the intersection of these arcs is connected (though possibly empty).
Then we have an embedding γ :

Hence by Lemma 4.2 one can find an open foliated embeddinḡ
One easily checks that ψ is an open foliated embedding which coincides with ψ 1 on (a−ε, b]×0 and with Notice that J can be covered by finitely many foliated charts contained in W . Lemma 4.3 allows to replace two consecutive foliated charts with one. Hence the proof follows from that lemma by induction on the number of foliated charts covering J. Cross sections. The following two lemmas describe general properties of cross sections.  ψ (a, b) × B n , P u = ψ (a, b) × u , u ∈ B n , be a plaque of ψ, and γ : V → X be a cross section. Then the following statements hold true.
(1) Suppose P u ∩ γ(V ) = ∅ for each u ∈ B n . Then then for each s ∈ (a, b) the restriction map ψ| {s}×B n : {s} × B n → X (4.1) is a cross section of ∆.
(2) Suppose γ(v) ∈ P u for some u ∈ B n and v ∈ V . Then there exists an open neighborhood In particular, the restriction ψ| s×Wv : s × W v → X is a cross sections of ∆.
(3) For every x ∈ S(γ(V )) there exist an open subset W of R n and a cross-section ψ Since γ(V ) intersects each leaf of ∆ in at most one point, it follows that distinct plaques P u and P v for u = v ∈ B n belong to distinct leaves of ∆. As B n is an open subset of R n , the map (4.1) is a cross section for each s ∈ (a, b).
(2) Consider the following map: where π is the standard projection to the second coordinate.
Then the assumption γ(v) ∈ P u for some u ∈ B n and v ∈ V implies that v ∈ γ −1 (U ) and ξ(v) = u.
Since the images of distinct points of V under γ are contained in distinct leaves of ∆, they also belong to distinct plaques of ψ, whence ξ is an injective continuous map between open subsets of R n . Hence, by Brouwer theorem on domain invariance ξ is an open map, [23]. In particular, ξ yields a homeomorphism of some open neighborhood (3) Let ω be the leaf containing x and y = γ(v) = ω ∩ γ(V ). If x = y, then one can put W x = V and γ x = γ.
(c) Let x ∈ S(ω 0 ) \ ω 0 . Then decreasing B n one can assume that the image of ψ does not intersect ω 0 , whence x / ∈ ω 0 . From arbitrariness of x ∈ S(ω 0 ) we conclude that ω 0 is closed in X.
Parallel cross sections. Let γ : V → X be a cross section and W ⊂ V be an open subset. Then a cross section δ : W → X parametrically agrees with γ, whenever for each u ∈ W the points δ(u) and γ(u) belong to the same leaf. Also δ is parallel to γ if it parametrically agrees with γ and δ(W ) ∩ γ(W ) = ∅.
Let γ 0 , γ 1 : V → X be two parallel cross sections intersecting only non-compact leaves. For each u ∈ V let ω u be the leaf containing γ 0 (u) and γ 1 (u), I u ⊂ ω u be the compact segment with ends γ 0 (u) and γ 1 (u), and Int I u be the interior of I u . In this situation we will put: Lemma 4.7. There exists a homeomorphism ψ : for every u ∈ V , see  Proof. Fix some ε > 0 and denote J = (−ε, 1 + ε). Then it follows from Corollary 4.4 and Lemma 4.2 that for each u ∈ V there exists a neighborhood W u in V and an open foliated embedding ψ u : J × W u → X having the following properties: In particular, this implies that the set As V is paracompact, there is a locally finite cover {W i } i∈Λ of V and for each i ∈ Λ an open foliated embedding ψ i : J × W i → X such that ψ i ([0, 1] × u) = I u for all u ∈ W i . Denote U i = ψ i (J × W i ) and U = ∪ i∈Λ U i . Then U is an open neighborhood of K(γ 0 , γ 1 ) and {U i } i∈Λ is a locally finite cover of U .
Let {λ i : V → [0, 1]} i∈Λ be a partition of unity subordinated to the cover {W i } i∈Λ . Thus supp(λ i ) ⊂ W i and i∈Λ λ i (u) = 1. Let also p i : J × W i → J and q i : J × W i → W i be the standard projections, and Then supp(µ i ) = J × supp(λ i ), whence µ i extends by zero to a continuous function on all of U .
Let f : U → J be the function defined by the following rule: Since for each u ∈ W i the function p i • ψ −1 i : I u → [0, 1] is homeomorphism which maps γ 0 (u) and γ 1 (u) to 0 and 1 respectively, and j∈Λ µ j ≡ 1, we see that the restriction f | Iu is a convex linear combination of orientation preserving homeomorphisms. Therefore f | Iu : I u → [0, 1] is a homeomorphism as well.
Let also g : U → V be the map defined by g(x) = q i (x) whenever x ∈ U i . Due to (b) this definition does not depend on a particular U i containing x. Hence g is a well-defined continuous map.
Then the mapping is a continuous bijection being also a local homeomorphism, and so it is a homeomorphism. Moreover, φ(I u ) = [0, 1]×u for all u ∈ V . Therefore ψ = φ −1 is the required homeomorphism.
Lemma 4.8. Let γ i : V → X, i ∈ Z, be a family of pairwise parallel cross sections intersecting only non-compact leaves and U = S(γ i (V ))) be the common saturation of their images. Suppose also that the following two conditions hold: Then U is open in X and foliated homeomorphic with R × V .
Proof. By Lemma 4.7 for each i ∈ Z there exists a homeomorphism is a segment of the leaf of ∆ between the points γ i (u) and γ i+1 (u); Therefore we have a homeomorphism

Proof of Theorem 2.14
Let γ : V → X be a cross section intersecting only leaves being simultaneously noncompact, properly embedded, and non-special. We have to prove that its saturation S(γ(V )) is open and foliated homeomorphic with R × V .
First we will assume that ∂X = ∅. The proof of the case ∂X = ∅ will follow from the case ∂X = ∅ by passing to the double 2X of X and considering the one-dimensional foliation on 2X induced by ∆. It will be given at the end op this section.
Our proof is based on the following statement which will be proved below.
Proposition 5.1. Let K ⊂ X be a compact subset. Then one can find two parallel cross sections α, β : V → X parametrically agreeing with γ and satisfying Moreover, if A, B : V → X are two parallel cross sections parametrically agreeing with γ, then one can assume that Before proving Theorem 2.14 let us deduce it from Proposition 5.1. Fix any increasing sequence K 1 ⊂ K 2 ⊂ · · · of compact subsets of X such that X = ∪ i∈N K i .
Using Proposition 5.1 one constructs a family of parallel cross sections α i , β i : V → X, i ∈ N, parametrically agreeing with γ and such that Exchanging α i and β i if necessary and re-denoting them as follows: γ −i = α i , and γ i−1 = β i for i ∈ N, one can assume that the sequence of cross sections {γ i } i∈Z satisfies assumptions of  The following lemma guarantees existence of local cross sections in Proposition 5.1.
Lemma 5.2. Let K ⊂ X be a compact subset. Then for each u ∈ V one can find an open neighborhood W in V and two parallel cross sections α, β : W → X parametrically agreeing with γ and such that S(γ(W )) ∩ K ⊂ L(α, β).
Proof. Suppose that lemma fails, so there exists u ∈ V belonging to some leaf ω such that • for any decreasing sequence W i of neighborhoods of u in V with ∩ i∈N W i = {u} • and any family of pairs of parallel cross sections α i , β i : W i → X, i ∈ N, parametrically agreeing with γ the set Then one can assume, in addition, that the following properties hold: (a) the sequence {x i } i∈N converges to some point x ∈ K; , whence x ∈ U as well, and so x ∈ ω. Indeed, (a) follows from compactness of K.
To prove (b) fix any continuous bijection φ : R → ω. By assumption φ is proper, so one can find A > 0 such that ω ∩ K ⊂ φ(−A, A). Choose α i and β i so that Finally, to satisfy (c) choose W i+1 so small that x i ∈ S(γ(W i+1 )) for all i ∈ N. Now let ω i be the leaf of ∆ containing x i , and y i = ω i ∩ γ(W 1 ). Then the sequence {y i } i∈N converges to y = γ(u) ∈ ω. Hence p(x) = p(y) = p(ω), while p(x i ) = p(y i ) = p(ω i ) for all i ∈ N. Therefore by Lemma 2.6 ω is a special leaf which contradicts to the assumption.
The rest of the proof of Theorem 2.14 is based on E. Michael's result about selections, [31]. Let 2 X be the set of all subsets of X and E(X) ⊂ 2 X be the set of all closed subsets of X. Let also A ⊂ V be a subset and q : V ⇒ X be a multivalued map, i.e. a map q : V → 2 X . Then a selection for the restriction q| A is a continuous map φ : A → X such that φ(x) ∈ q(x) for all x ∈ A.
A multivalued map q : V ⇒ X is called lower semi-continuous whenever for each open U ⊂ X the set A family Z ⊂ 2 X is called equi-LC k , k ≥ 0, if for every P ∈ Z, x ∈ P , and a neighborhood U x of x in X, there exists a neighborhood O x of x in X such that for every Q ∈ Z every continuous map f : A topological space Z is called C k , or k-connected , k ≥ 0, if every continuous map f : S m → Z of an m-sphere (m ≤ k) is homotopic to a constant map.
Theorem 5.3. [31, Theorem 1.2] Let V be a separable metric space, A ⊂ V be a closed subset with dim(V \ A) ≤ k + 1, X a complete metric space, Z ⊂ E(X) be equi-LC k and q : V → Z be a lower semi-continuous map. Then every selection for q| A can be extended to a selection for q| U for some open U ⊃ A. If also every S ⊂ Z is C k , then one can take U = X.
We will use the following particular case of Theorem 5.3.
Corollary 5.4. Let V be a separable metric space, dim V = n, X be a complete metric space, and Z ⊂ E(X) be an equi-LC n+1 family such that each Q ∈ Z is contractible. Then every lower semi-continuous multivalued map q : V → Z has a continuous selection.
Proof of Proposition 5.1 Since V is paracompact, it follows from Lemma 5.2 that there exist • a locally finite open cover W = {W i } i∈N of V with compact closures W i , and • a family of pairs of parallel cross sections α i , β i : W i → X, i ∈ N, parametrically agreeing with γ such that whenever the cross sections A, B : V → X are given, and Then it follows from Lemmas 4.7 and 4.1 that for each i ∈ N one can find an embedding Let u ∈ V , ω u be the leaf of ∆ containing γ(u), and φ u : R → ω u be any bijection satisfying φ −1 u (α i (u)) < 0, and φ x (0) = γ(u). Therefore φ −1 u (β i (x)) > 0 for all i such that u ∈ W i . Then there are two numbers a u , b u such that consists of two half closed intervals A u = φ u (−∞, a u ] and B u = φ u [b u , +∞).
Since ω u is a properly embedded leaf, it follows that A u and B u are closed in X. Moreover, by (3) they do not intersect K.
Define the following two maps a, b : V → E(X), i.e. multivalued mappings a, b : V ⇒ X with closed images, by a(u) = A u , b(u) = B u for u ∈ V . Let u ∈ V be such that A u ∩ U = ∅, and x ∈ A u ∩ U . Since U is open, one can assume that x is not the end of A u , that is φ −1 u (y) < a u . By assumption u ∈ W i for some i ∈ N. Then by (ii) Notice that for each x ∈ X there exists an open neighborhood U x such that the intersection of U x with each leaf ω is either empty or homeomorphic to an open interval. Therefore intersection of U x with each set A u is either empty or homeomorphic to (0, 1) or to (0, 1]. In the latter two cases U x ∩ A u is contractible. Hence every continuous map S k → U x ∩ A u is null homotopic and one can put O y = U y . This means that A is equi-LC k for all k ≥ 0.
Since for each u ∈ V the sets A u and B u are contractible, it follows from Lemma 5.5 that a and b satisfy assumptions of Corollary 5.4. Hence they admit continuous selections α, β : V → X and these selections are the required cross sections. This completes Proposition 5.1.
Proof of Theorem 2.14. Case ∂X = ∅. We need the following simple lemma whose proof we leave for the reader. be the double of X, i.e. the union of two copies X 1 and X 2 of X glued along their boundaries by the identity map. Let also σ : X → X be the involution interchanging X 1 and X 2 by the identity map. Then the foliation ∆ on each of the copies of X gives a one-dimensional foliation ∆ on X. Moreover, let V be the double of V as in Lemma 5.6. Then V is open in R n and the cross section γ naturally extends to the cross section γ : V → X of ∆ such that γ| V = γ and σ • γ = γ • ξ.
Since ∂ X = ∅, it follows from the boundary-less case of Theorem 2.14 that the saturation S( γ( V )) is open in X and foliated homeomorphic with R × V . That homeomorphism induces a homeomorphism of the open subset S(γ(V )) = S( γ( V )) ∩ X 1 of X 1 onto R × V . Theorem 2.14 is completed.

Acknowledgments
The authors are sincerely grateful to Olena Karlova for pointing out to Katetov-Tong theorem about characterization of normal spaces which leads us to the proof of Theorem 2.14 using E. Michael selection theorem.