NONSEPARATED MANIFOLDS AND COMPLETELY UNSTABLE FLOWS

We define an order structure on a nonseparated n-manifold. Here, a nonseparated manifold denotes any topological space that is locally Euclidean and has a countable basis; the usual Hausdorff separation property is not required. Our result is that an ordered nonseparated n-manifold X can be realized as an ordered orbit space of a completely unstable continuous flow on a Hausdorff (n + 1)manifold E.


INTRODUCTION
Nonseparated manifolds arise in a very natural way in the study of ordinary differential equations and completely unstable flows.A topological space that is non-Hausdorff, locally Euclidean and has a countable basis is referred to as a nonseparated manifold.A flow on a manifold E is said to be completely unstable if it has no nonwandering points.Such systems occur very naturally.For example, 2 on any continuous flow without equilibria is completely unstable, and the restriction of any flow to the complement of its set of nonwandering points is completely unstable.All open manifolds admit completely unstable flows.Let E x E be a completely unstable c o flow on an (n + 1)-manifold E.
The orbit space of is the set E/ of all orbits of with the quotient topology (the finest topology in which the natural projection E E/@ is continuous).If admits local cross-sections at every point of E that are n-Euclidean, we say @ is locally trivial.It is known that if either E and are c or n 2, then is locally trivial ([1], [2]).Moreover, if is locally trivial, completely unstable c o flow then E/ is a nonseparated n-manifold.The ordered orbit space of is obtained from this non-separated manifold by imposing an additional structure that indicates the order in which the cross-sections of @ that correspond to the charts of E/ are traversed by orbits of (precise definitions are given in [3]).We then have the following classification theorem which shows that completely unstable flows on manifolds can be classified completely in terms of their associated ordered orbit spaces (Theorem 3.1, [3]).

CLASSIFICATION THEOREM If
and ' are locally trivial, completely unstable o c flows on m-manifolds M and M', respectively, then (M,) and (M',') are topologi- cally equivalent if and only if M/ is order isomorphic to M'/'.Our interest here is in the question of realization" What nonseparated mani- folds can be realized as the ordered orbit space of a completely unstable c o flow on some Hausdorff manifold?Some restrictiorl on the nonseparated manifold is undoub- tedly necessary.However, it appears to be a difficult problem to characterize the realizable ones.We present a preliminary result in this direction in the present paper.We first define a restricted class of nicely ordered nonseparated manifolds.We then prove that these manifolds are all realizable.
REALIZATION THEOREM.If X is nicely ordered, nonseparated n-manifold then X can be realized as the ordered orbit space of a completely unstable continuous flow (E,), where E is a Hausdorff (n + 1)-manifold.
Essentially the same result, in the case X is a one-dimensional simply connec- ted variety and E =jR 2 is stated in Haefliger and Reeb [4].It is also stated in Neumann [3] for one-dimensional manifold X.
In {}2 below, we give most of the definitions and notation required in the proof of the realization theorem; the proof itself occupies {}3 {}5.Finally, in {}6 we prove the following corollary.
COROLLARY.Let X and E be as in the realization theorem.If n(X) O, then n (E) 0 for n > 1.Moreover, if X is a one-dimensional simply connected nonsepar- 2 ated manifold, then E is homeomorphic to m 2. PRELIMINARIES.

DEFINITIONS AND NOTATION.
Throughout what follows, E denotes a Hausdorff (n + 1)-manifold, E x jR1 E denotes a continuous flow on E, and X denotes a nonseparated n-manifold with a countable basis (V i, v i) where each V is homeo- morphic to Dn, the compact unit n-disk.A topological space is a nonseparated manifold if it is locally Euclidean and has a countable basis, the usual Hausdorff separation axiom is not assumed For any set S C_E, T c_ m 1 S. T toT}; x-T {x}-T; and for xcE and tcIR I, x-t Ct(x) (x, t).The orbit of xcE is the set y(x) x .]1.The orbit space E/ is the set of all orbits of with the quotient topology.Also, throughout what follows, for any set A contained in a topological space, and X will denote the interior and the closure of A respec- tively.
A set U E is said to be wandering (with respect to ) if there exists 1 t o tim such that U.tFU for each t with It12 t o A point xcE is nonwandering if it has no wandering neighborhood.Equivalently, xcE is nonwandering if xcJ+(x), here J+(x) denotes the set of limits of sequences {x n tn}, where {x n} converges to x and t n tends to (R).The (closed invariant) set of all nonwandering points of is denoted as f().A flow is said to be completely unstable if () .A 1 cross-section of is a set S c_ E for which the mapping h'S x m E defined by h(s,t) s.t is a homeomorphism of S x m l onto a subset of E.

STATEMENT OF THE REALIZATION THEOREM.
In order to state our main result, we first need to define the following order structure.DEFINITION 3.1.Let X be a nonseparated n-manifold with a countable atlas Vi' i where each V is homeomorphic to D n (the compact unit n-disk), and o {Vi}i> forms an open cover for X.We say that X is nicely ordered if there exists a collection of continuous functions hij'ViC Vj {-I,I} satisfying: (a) hij(x) -hji(x) for every xViC Vj; (b) If xV C Vj V k with hij(x) +1(-1) and hjk(X +1(-1), then hik(X) +1(-1) (c) If {x n} is a sequence in V Vj V xV i, and xVj, then xV k.

k (i<j<k) converging to
The order structure defined as above is a generalization of the order structure on a nonseparated 1-manifold, as given by Neumann in [5].However, the property (c) of the order structure as above is slightly more restrictive than the property (3) of the order structure given by Neumann (see 3.5 below), and thus the phrase "nicely ordered" is used.Our main result is the following Realization Theorem.
REALIZATION THEOREM.3.2.If X is a nicely ordered, nonseparated n-manifold, then X can be realized as the ordered orbit space of a completely unstable continu- ous flow (E,), where E is a Hausdorff (n + I) -manifold.
REMARKS 3.3.() This result in the case X is a one-dimensional simply connected variety and E =JR 2 is stated in Haefliger and Reeb [4].It is also stated in Neumann [3] for one-dimensional manifold X.
(B) Properties (a) and (b) of the order structure defined in 3.1 above will be used implicitly throughout the proof of the realization theorem.OUTLINE 3.4.We shall prove the realization theorem by induction on the number of charts in X in the following two steps.
(1) We first show that X can be realized as a base space of a bundle B < E,p,X >, where E is a Hausdorff (n + 1) -manifold.(2) We then define a flow on E, show that it is completely unstable and finally show that X is the orbit space of the dynamical system (E,).
The first step, that is to show the existence of the bundle B= < E,p,X >, is the major step in the proof of the realization theorem.DISCUSSION 3.5.We would like to point out that the direct generalization of the order structure given by Neumann for nonseparated 1-manifold in [5] would be: (a) and (b) same as in the definition 3.1 above and replace (c) by a less restric- tive condition (c') as follows: and hjk(xn) for each n, and (B) x n/ xV with xVj, then x V k.
Moreover, if is a completely unstable c o flow on the (n + 1) manifold E and admits cross-sections that are locally Euclidean, then E/ can be ordered in this sense" choose a covering system {Si}i> of cross-sections for the dynamical system 0 (E,) (see 4.2, 4.3 of [3]).Set V P(Si) for each i.Then vi}i> forms an open be defined as in the proof of the classifi- cover of E/.Let fij" Vi Vj , cation theorem (Theorem 3.1, [3]).Set hij(x) sgn(fij(x)) xV Vj.Using the properties of fij (see [3]), it is now immediate that hij satisfy the properties (a), (b) and (c') above.

4.
EXISTENCE OF A BUNDLE B < E, p, X >.
In the setting of the existence theorem (Theorem 3.2 of [6]), to show the existence of a bundle B=<E, p, X >, we seek the coordinate transformations {gij} in the space X, with the structure group the group T of all translations of ml.In particular, we seek the maps: gij" Vi f Vj T satisfying" (a) gij(x) ogjk(X) gik(X) for each x V C VjC V k (compatibility c ond t on (b) If {xn} is a sequence in ViNVj such that {xn} converges to both x V and xj e Vj with x xj, and hij(xn) +1(-1) for all n, then gij(xn)(t) + (-) as n (for every t il).
We define gij in terms of the translations fij as follows" gij" iC Vj and t m Where fij" ViF') Vj IR are to be defined so as to satisfy: (A) fij(x) + fjk(X) fik(X) for each x ViF VjC Vk; and (B) If {xn} is a sequence in X such that {xn} converges to both xi i and xj j with x i p xj, and hij(xn) +I(-1) for all n, then fij(x n) + (-) as n .
Thus, to show the existence of the coordinate transformations {gij}, we need to show the existence of the translations {fi_i satisfying (A) and (B) above.We show the existence of {fij by induction on the number of charts in X.Note that since each chart is Hausdorff and X is not, X can not have a single chart.
REMARK 4.1.One should note that the existence theorem (Theorem 3.2 of [6]) not only gives the existence of a bundle B < E, p, X > but also its uniqueness up to bundle equivalence.
NOTATION 4.2.In What follows, B 1 denotes the set of all non-Hausdorff points of X and Vij(i # j) denotes the set of all those points x B 1C V such that there exists a sequence {x n} in V Vj with {xn} converging to x and also to another point xj Vj with x P xj.Note that Vij is the set of all those non-Hausdorff points in V that can not be separated from some point in Vj.PROPOSITION 4.3 For any # j, the set Vij is a closed subset of the metric space Vi.
PROOF" Let {yk be a sequence in Vij such that {yk converges to y c V i.We want to show that y c Vij. Without loss of generality, let Yk eB (y) for each k, where B 1 (y) is an open ball in the metric space v i Moreover, for each k, let {x} be a sequence in V C Vj such that {x} converges to Yk and also to another point y Vj, with Yk # Y" Since Vj is compact, the sequence {y} has a convergent subsequence {y} /y' Vj.As above, let y B{__(Y') for each converges to both y and y'.Moreover, it can be easily seen that y # y'.Hence EXISTENCE OF fij FOR TWO CHARTS 4.4.If X has only two charts, say V and V 2, then {fij} (I < i, j < 2) satisfying (A) and (B) above exist for these two charts.
Let X' (V 11] V 2) I V12 (disjoint union).Note that X' is a metric subspace of the metric space V I. Define f" X' [0,1] by f'(x) Then f' is a continuous function, and since V12 is a closed set (4.3), f'(x) 1 if and only if x V12.Define f2" X' [0,(R)] by f2(x) tan# (f'(x)).Then f2is continuous and f2 (x)   if and only if x V12.Now set f12 f12 Vlr V 2, where f121V1/ V 2 indicates the restriction of the function f12 on the set V 1f V 2.
Finally, the set f12' f21 -f12' and fii O(i 1,2) is the desired set of f.. (1 < i, j < 2), satisfying (A) and (B) above.This completes the construction of fij in the case X has only two charts.
REMARK 4.5.In the construction of f12 above, observe that f12(x) > 0 for every x VIF V 2. In the rest of the proof, we would construct fij so as to s'atisfy (A) and (B) above and also the following added property" (C) If j i, then fij (x) > 0 for every x V FI Vj.
INDUCTION STEP 4.6.Suppose that we can define {fij} (1i, j<n) satisfying (A), (B), and (C)above in the case X has n-charts say V 1, V 2, V 3 V n, we show that {fij satisfying (A), (B), and (C) above can be defined in the case X has (n + I) -charts V 1, V 2 V n, Vn+I.
In order to show the existence of {fij} in the case X has (n + 1) charts, we first need to show the existence of {fij} in the case X has only three charts, which in turn requires the following lemma" LEMMA 4.7.Let A and B be closed subsets of a metric space Y.If g-A [0,1] is a continuous map such that g(x) 1 if and only if x AB, then g can be extended to a continuous map " Y [0,1] such that (x) 1 if and only if x B.
It is obvious that gALpB is a well-defined map that extends g.Also, it is contin- uous by glueing lemma ([7], page 50).Moreover, gAtPB(X) if and only if x B.
In order to extend gAUB to the whole of Y, we observe that A LB is a closed subspace of the metric space Y. Therefore, there exists a continuous function u'Y [0,1] such that u(x) if and only if x AU B.Moreover, by Tietze Extension Theorem, there exists a continuous extension g'-Y [0,1] of gAU B such that g'(x) if x B.
Finally, define " Y [0,I] by (x) u(x) g'(x) for x Y.It can be easily seen that is the desired map.This completes the proof of the lemma.
EXISTENCE OF fii FOR THREE CHARTS 4.8.If X has only three charts, say V 1, V 2, and V 3, then {fij} (1<i, j<3) satisfying (A), (B) and (C) above can be defined for these three charts.
Let f2" (V1( V2)L] V12 [0, ] be the function as obtained in the case of two charts (cf.4.4).Define f3" (V2CV3)LJ V23 [0, ] analogous to f2" Here, V23 is a set as defined in 4.2.In order to define f3" (Vl C V3)L] V13 [0, ], let A123 V 1C V2cV3 and define V123 to be the set of all those points x BIC V 1, such that there exists a sequence {x n} in A123 with {x n} converging to x I and also to another point x 3 in V 3 with x # x 3, (B 1 is the set as defined in 4.2 above).Observe that V123C_ V13.
We claim" THEOREM 4.9.If x V123 then either x V12 or x V23.
PROOF" If x V123 then there exists a sequence {x n} in A123 such that {xn} converges to x and also to another point x 3 in V 3 with x # x 3. Since V 2 is compact, {x n} has a convergent subsequence {x k} x 2 V 2. If x 2 x I # x 3, then x 1 V23, otherwise x I V12.We now define f{3" A123 U V123 [0, -] by f3(x) f2(x) + f3(x); x A123U V123.Then f13 is a well-defined continuous map Since both f12 f23 A123, it follows from 4.9 above that f3(x) if and only if x V123.We want to extend f3 continuously to f3" (VlC V3) U V13 [0, (R)] such that f13(x) if and only if x V13.(Note that the extension of f3 is also denoted as f3 ).
In the setting of the lemma 4.7 above, we have Y (VIN V3) U V13, A A123 U V123 and B V13. Assuming A to be a closed subset (proved below) of Y, define g-A [0,1] by g(x) 2_ arc tan (f3(x)) where f3 is defined by (4.2) above.Let -Y [0,1] be an extension of g as obtained in the lemma 4.7.Define f13 Y/[O x To com- (R)] by f13 tan ((x)).Note that f3(x) if and only if x B VI3. plete the definition of f3' we still need to show: THEOREM 4.10 The set A A123U V123 is a closed subset of Y (VlFV3)U V13.
PROOF" As in proposition 4.3, it can be seen that V123 is a closed subset of Y. Thus, to complete the proof, it suffices to show that the closure of A123 in Y is contained in A.
Let {xn} be a sequence in A123 such that {xn} x I Y. Since Y (V 1N V 3) U V13 (disjoint union), either x I V 1 C V 3 or x I V13.Let us first consider the case x V 1N V 3. Since V 2 is compact, {x n} has a convergent subse- quence {x k} x 2 V 2. We claim that x 2 V and thus x 2 x 1.If not, then x I V 2, Thus by property (c) of the definition of order structure, we have x I V 3, a contradiction.Hence, in his case, x A123A.
If x V13, then x # V 3. Also, since V 3 is compact, therefore the sequence {x n} admits a convergent subsequence {x k} x 3 V 3. Hence x V123 c_A, as des red.
Finally, f13 fi31V1FV3' f12 f21VlF) V 2, f23 f31V2 V3' fiiv -fiJ (1 i, j < 3) and fii O(i 1,2,3) is the desired set of {fii} satisfy- ing (A), (B) and (C) above Here, fjlV i. F V i denotes the restriction of fi. on V 0 Vj(I < i, j 3).This completes the construction of fii in the case X has three charts We now return to our induction step.We want to define {fij} (1 < i, j < n+l) satisfying (A), (B) and (C) in the case X has (n + I) charts V I, V 2 Vn+I, knowing that {fij} (I-< i, j < n) satisfying (A), (B) and (C) have already been defined in the case X has n charts V 1, V 2 V n.For convenience sake, we will use the following notation in the rest of the proof.NOTATION 4.11.Any extension of fj will be denoted as fj.For any i, j and k, Aij k denotes the set V iF VjC V k, and Vij k denotes the set of all points x in 6' 1C V (B is the set of all non-Hausdorff points in X) such that there exists a sequence {xn} in Aij k with {x n} converging to x and also to another point x k in V k with x x k.Moreover, for any # j, Aij denotes the set V C Vj. REMARK 4.12.For any i<j<k, the set AijkU Vij k is a closed set in AikLJ Vik (cf.4.10) and it would be denoted as Bijk.This remark would be used implicitly throughout what follows.
We now start defining fij for (n + 1) charts.Define f'n n+l" An n+lUVn n+1 [0,-] as in the case of two charts (cf 4 4) Next define f'n_l n+1" Bn-1 n n+1 [0, (R)] by f'n-1 n+l (x) f-I n (x) + f' (x); x Bn_ n n+l 1 n n+l (4.3) as in 4 9 for three charts.Here f' has been defined at the induction step.n-1 n Using lemma 4 7, extend f' n-1 n+1 to fn-1 n+l An-1 n/1 -) Vn-1 n+1 [0,(R)] as was done in the case of three charts.We next define the function f' n-2 n+l as follows where f' and f' have been defined at the induction step and f' n-2 n n-2 n-1 n-1 n+l is obtained above.Using the induction hypothesis and (4.3) above, it can be easily seen that f' is well defined that is f' defined by (4 4) coincides n-2 n+l n-2 n+l with fn-2 n+l defined by (4.5) on the intersection (Bn_ 2 n n+l ' Bn-2 n-1 n+l )" Finally, using lemma 4.7 with Y An_ 2 n+l ) Vn-2 n+l' A Bn_ 2 n n+lBn-2 n-I n+l, and B Vn_ 2 n+l extend f' n-2 n+1 to f.2 n+l An-2 n+llJVn-2 n+l done in the case of three charts.
[0,(R)] as was Continuing this process we obtain f' f' n-3 n+l' n-4 n+l' and f n+l tively.Finally, define f n+l as follows" induc- ,n) are the functions obtained above and f'lj (J 2,3,   n) are the functions that have been defined at the induction step.Using the induction hypothesis and the definition of the functions f! n+1 (2<i<n) it can be seen that f n+1 is well defined.Finally, using lemma 4.7 with Y A 1 n+IL)V1 n+l' n A iU=2(Bli n+l and B V I n+l' extend f n+l to f n+l A1 n+lLPVl n+l +[o,(R)] as was done in the case of three charts.
We claim that the set {fij}(1 < i, j < n + I) so obtained is the desired set of functions satisfying (A), (B) and (C).From the construction of {fij} it is obvious that the functions {fij} satisfy both (B) and (C).For (A), we need to show that fij(x) + fjk(X) fik(X) for each x V C Vj V k and for any i, j and k where 1 <i, j, k < n + 1.In view of induction hypothesis, we only need to prove it in the case when one of the i, j or k is n + 1.
The cases when j or k equals (n + I) are analogous.This completes the induction step and hence the construction of {fij} for i, jl.
Hence, by the existence theorem (Theorem 3.2, [6]), we get a bundle B <E,p,X> with the base space X and the coordinate transformations {gij}.Also, any two such bundles are equivalent.Moreover, since X is an n-manifold, E is an (n + 1) manifold.We finally show that THEOREM 4.13 E is a Hausdorff space o PROOF: If not, let e and e' be two nonseparated points in E. We have {Vk}ka I covers X, and since for each k there exists a homeomorhism k" Vk x 1 p-l(Vk) ([6], page 7), each p-l(v k) is Hausdorff.Consequently, there exists j > i, such that e p-l(v i) and e" p-l(vj) with p-l(vi) p-1(Vj) # B.Moreover, there exist x I' x' jand t, t' m I, such that i(x, t) e and j(x',t') e'.Let n {vn.} and {V} be neighborhood systems at e and e', respectively, with V n>l n>l o p_l(vi )o and V n .J C_ p-I(vj) for all n.Since e and e' are nonseparated points, there- n n fore, for each n, there lexists Yn Vi ( Vj.Let Xn P(Yn Vi Vj for each n. Then there exists t n ]R such that Yn i(Xn'tn 'j(Xn'gji(Xn)(tn )) for each n, where gij are the coordinate transformations as constructed above.
Since Yn converges to both e i(x' t) and e' j(x',t') and both i and j are homeomorphisms, it can be easily seen that x n converges to both x and x'; t n t;l and gji(Xn)(tn) t' as n .Since tn t, therefore there exists < t o for all n.Consequently t o IR such that t n gji(Xn)(tn) gji(Xn)(to) for all n.

X AS AN ORDERED ORBIT SPACE.
We now show that we can define a completely unstable continuous flow on E and that X can be realized as an ordered orbit space of the dynamical system (E, ), where E is the Hausdorff manifold obtained in {} 4 above.m To define a flow @-E x E. Fix (q, s) E x Since v.i'Vj x -1 p (Vj) is a homeomorphism for each j and {Vj} cover X; therefore, there exists j>l some k > with x Vkand t such that q Vk(X, t).Define (q, s) k(X, t + s). (5.1) We first show that is well defined; that is, if q also equals j(x',t') for some j # k, x' V i and t' m I, then k(X, t + s) j(x', t' + s).
We next show that is a continuous flow.It is obvious that is a continuous function.Moreover, satisfies the group law for (q,0) k(x,t + 0) q; and ((q,sl),S2) (k(x,t + Sl), s 2) k(X, t + s I + s 2) (x, s + s2).
We finally show that is completely unstable and that X is the orbit space of the dynamical system (E, @).To show that is completely unstable, fix q E. We want to show that q admits a wandering neighborhood.Let q k(X, t o for some x

2 where
B{ (y') is an open ball in Vj.By induction, there exists N m > y)B1 (y)" Now the sequence {x m} is in Vi2/Vj and obviously 2 m 2 m in z + such that y Vij as desired.