NEW ZEALAND JOURNAL OF MATHEMATICS Volume 28 (1999), 285-297

DOMAINS OF PERFECT LOCAL HOMEOMORPHISMS

M a t h e w Tim m (Received May 1997)

Abstract. The structure of domains of perfect local homeomorphisms is inves­ tigated. Finiteness conditions for domains of perfect local homeomorphisms are presented. A sufficient condition for all components of the domain of a perfect local homeomorphism to be open is given.

1. In tro d u c tio n

In this paper all maps f : X —+ Y between topological spaces X and Y are continuous functions. All one point sets are closed and a neighborhood about x is an open set that contains x. A map / : X —> Y is a local homeomorphism if for each x € X there is a neighborhood Ux about x and a neighborhood Vx about y = f(x) such that the restriction ( / | Ux) : Ux —► Vx is a surjective homeomorphism. A map f : X —> Y is proper, or compact, if and only if, given any compact subset K c Y, it follows that f~ 1(K) is compact. A map p : X —> Y is perfect if and only if, for every y € Y, p~l (y) is compact. Note that it is not assumed that perfect maps are closed, even though this seems to be fairly common in the literature. The cardinality of a set S is denoted by |5|. The main result of Jungck and Timm [3], stated below, is a decomposition theorem for the domain of a proper surjective local homeomorphism. It is a gen­ eralization of Jungck’s result [2, 2.1] that is a decomposition theorem for point inverses under proper local homeomorphisms.

Theorem 1.1 ([3, 1.4]).Let X and Y be Hausdorff. Assume that Y is connected. If p : X —> Y is a proper, surjective, local homeomorphism then X = Xo U ... U X n for some n >0, with each X ja component of X . Furthermore (p \ X j): X j —► Y is a proper, surjective, local homeomorphism.

The proof of the above theorem, together with the collection of examples we had at the time, prompted the question (stated as Question 2.3 in Jungck and Timm [3]) of whether it was necessary to assume that a surjective local homeomorphism p : X —> Y be proper, in order to assure that every component of X is open in X. In addition, one is struck by the fact that both the result from Jungck [2, 2.1] and that from Jungck and Timm [3, 1.4] can be thought of as giving finiteness conditions on domains of proper local homeomorphisms. Recent attempts to answer Question 2.3 [3], as well as an attempt to gain a deeper understanding of the ideas presented in [3] have generated the questions below.

1991 AMS Mathematics Subject Classification: Primary 54C10; Secondary 54E40, 54F15. Key words and phrases: perfect map, proper map, compact map, finiteness conditions. 286 MATHEW TIMM

Question 1.2. Given a map p : X —► Y, what conditions on p, X , or Y (other than the obvious one) will assure that the domain X is a union of finitely many components? In particular, to what degree can the requirement of [3, 1.4], thatp be proper, be relaxed? Specifically, ifp : X —> Y is a perfect local homeomorphism, what sorts of finiteness conditions can be obtained on Y? If one cannot find reason­ able conditions under which a non-proper local homeomorphism’s domain must be a union of at most finitely many components, what can be said about the domain in this situation? This paper presents a collection of interesting examples and results suggested by these examples. In Section 2, what is known about relationships between proper, perfect, and closed are summarized. In Section 3, the finiteness conditions onX that follow from the assumption that p : X —► Y is a perfect local homeomorphism are studied. It is in this context that most of the examples are presented. Also in this context we develop analogs of the results [2, 2.1] and [3, 2.1] and present a solution to Question 2.3 [3] for perfect local homeomorphisms. Section 4 contains a brief investigation of what happens if one attempts to assume that the local homeomorphism p : X —» Y satisfies some sort of “local properness” condition. While I am most interested in the situation when p is a local homeomorphism, some of the results are stated more generally. The reader should compare the finiteness conditions obtained in this paper to those that can be obtained when one assumes that the spaces or maps have additional structure imposed upon them. In addition to the works mentioned above, a reasonable summary of these sorts of results can be found in McAuley and Robinson [4]. There are illustrations that accompany several of the examples. All the illustrations are at the end of the manuscript. I thank Jerry Jungck and Ollie Nanyes for our frequent conversations about the material in this paper. Jo Heath and Andrew Lelek are thanked for their readings of an earlier version of the paper. The referee is thanked for the care which (s)he took reading the manuscript and in particular for pointing out Example 3.15. Thanks to Mike McAsey and Alberto Delgado for their assistance with the Tex version of the manuscript. Finally, thanks to Morgan Rose for taking naps.

2. On Relationships Between Proper, Perfect, and Closed After an inspection of the proof of [3, 1.4] one is tempted to include the condi­ tion, as occurs frequently in the literature, that a perfect map be closed. But, if a map is perfect and closed, then it is proper, so this is too strong a requirement. It immediately places one in situations already studied in Jungck [2] and Jungck and Timm [3]. However, for the sake of completeness, this section include a brief treatment of what is known about the relationships between the three properties: perfect, closed and proper. The proofs of the results in this section are straight­ forward and are all similar. One of the proofs is included to illustrate the general method. Related results are contained in the papers by Ho [1] and Palais [5].

Lemma 2.1. Let p : X —> Y be a map. Then p is closed if and only if for each y E Y and each open subset U of X such that p-1(y) C U, there is an open V C Y such that y € V and p_1(y) C p-1(V) C U.

Lemma 2.2. Let p : X —*Y be a perfect closed map. Then p is proper. DOMAINS OF PERFECT LOCAL HOMEOMORPHISMS 287

Proof. Let K c Y be compact. Let U = {Ua : a E 1} be an open cover of p 1(K). Since p is perfect, for each y E Y, there is a finite collection Uayl, ... ,UayNy of U that covers p~1{y). Since p is closed we may apply Lemma 2.1 and for each y e Y obtain an open set Vy in X , such that p~1(y) C p~1{V) C Uk h ^ a yk- But K is compact, so there are finitely many t/’s, say yi, ■ ■ ■ , yM such that Vyi,... , VyM N " covers K. So it follows that U ^=1(Uj=iUay ) is a union of finitely many sets in U that covers p~x(K). Therefore, p~1(K) is compact. □

Note that if p : X —> Y is, in particular, a perfect closed local homeomorphism, then it is a proper local homeomorphism. It then follows, by Jungck [2, 2.7], that if X and Y are first countable Hausdorff spaces, Y is connected, and p : X —»Y is a surjective, perfect, closed local homeomorphism, then X is a finite sheeted covering space of Y with covering projection p. This section closes with a list of some of the relationships between the three concepts: perfect, closed, and proper. Recall that a Hausdorff space X is a k-space if and only if whenever F C X is such that for every compact K C X, K fl F is closed, it follows that F is closed. Following the fact are examples that elucidate these results.

Fact 2.3. Let p : X —> Y be a map. (a) If p is open (in particular, if p is a local homeomorphism) then p is proper if and only if it is perfect and closed.

(b) If X is first countable and Y is Hausdorff then p is proper if and only if it is perfect and closed.

(c) If Y is first countable and Hausdorff then p is proper if and only if it is perfect and closed.

(d) If X is compact and Y Hausdorff then p is closed. So, if p is perfect, then it is proper.

(e) (Palais [5]) IfY is a fc-space and p is proper, then p is closed. So, if Y is a fc-space, then p is proper if and only if it is perfect and closed.

Example 2.4. This example illustrates the necessity of the assumption that p be closed in Facts 2.3 (a) and (c) and the assumption that Y be Hausdorff in Fact 2.3 (b) and (d). Let Yq = {(0,0)}U{(l/n, : 0)n E N}. Lety* denote a fixed point not in Yo- Let Y = Yo U {y*}. The basic open sets about points z E Yo are those inherited from the standard topology on M2. The basic open sets about y* are all those of the form {y*} U {(1/n, 0) :n > no} for noE N. Now, let Xq = Yo where the topology is the subspace topology inherited from R 2. Let X\ = {(0, l)}U {(l/n , 1) E : n N} with the topology it inherits from R 2. Let X = XqU Xi. Note that X is first countable, Hausdorff and that Y is first countable but not Hausdorff. Now define p : X —* Y by p (l, 0) = y*, p(0,0) = (0,0), andp(l/n, m) — (1/n, 0), for m = 0,1. Note that p is a continuous, local homeomorphism from X to Y. It is perfect by Lemma 3.1, below. But, it is not closed since F = X\ is closed while p(F) = {y*} U {(1 /n ,0) : nE N} is not closed since it does not contain (0,0) which is a limit point of p(F). Further­ more, p is not a proper map since Yq is compact while p~1(Yo)= X\(0,1) is not. 288 MATHEW TIMM

This example also illustrates that the Hausdorff property is not a topological prop­ erty that is preserved by perfect, local homeomorphisms. E xam ple 2.5. Let Rcf denote the real line with the co-finite topology. Let / cf denote the open interval (0,1) as a subspace of Rcf. Let X be the disjoint union flcfU/cf. Define p : X —* Rc{ by p(x) = x. The map p is proper but it is neither open or closed since Icf is both open and closed in X yet p(Icf) is neither open or closed in Rc{- Note p is a locally one-to-one map that is not a local homeomorphism and that neither the domain or range of p is Hausdorff or first countable.

3. Finiteness Conditions on Domains of Perfect Maps Jungck’s result [2, 2.1] giving the finiteness condition on the cardinality of the inverse image of a point in the range says that if p : X —► Y is a proper surjective lo­ cal homeomorphism between first countable Hausdorff spaces and Y is connected, then there is a fixed n G N such that for all y G Y, |p—1 (y) | = n. That is, in this situation \p~1(y)\ is globally constant and finite on Y. So Theorem 1.1 and [2 , 2 .1] give, respectively, the global finiteness condition (that the domain consists of finitely many components) and the point-wise finiteness condition (that point in­ verses consist of finitely many points) for domains of proper local homeomorphisms between first countable Hausdorff spaces. The hope was to produce global and point-wise finiteness conditions for the domains of perfect local homeomorphisms. The first lemma is the analog of Jungck [2, 2.1]. The examples following it show that this is the best possible result. Lemma 3.1. Let p : X —>Y be a locally one-to-one map. Then p is perfect if and only if for each y G V, |p—1 (y) | is finite. So, a local homeomorphism p : X —> Y is perfect if and only if, |f>- 1(2/)| is finite for all y e 7

Proof. Let p : X —> Y be perfect. Suppose there is a y G Y, such that \p~l{y)\ is infinite. For each, x € p~1{y) there is an open set Ux such that x G Ux and the restriction (p \ Ux) : Ux —► p(Ux) is one-to-one. Since p~l (y) is compact, there are finitely many x, say x\,... , xn such that UXl,... , UXn cover p~1(y). Hence some UXi contains infinitely many points of p~1{y). Contradiction. The proof of the converse is immediate since any finite collection of points is compact in any topological space. □

Definition 3.2. Let p : X —* Y be a map. We say |p-1(y)| is locally constant at y G Y, if and only if there is an open set U about y and an integer n G N such that for all z G U, \p~l(z)\ = n. We say that \p~1(y)\ is locally constant on Y if and only if |p- 1 (f/)| is locally constant at every y G Y. We say |p—1 (y) | is locally bounded at y G Y, if and only if there is an open set U about y and an integer n G N such that for all z eU ,|p- 1 (z)| < n. |p- 1 (y)|is locally bounded on Y if and only if |p- 1 (y)| is locally bounded at each y GV.

So, while Jungck [2 , 2.1] gives that proper local homeomorphisms between first countable Hausdorff spaces are such that \p~1(y)\ is finite and globally constant, Lemma 3.1 only implies that for any perfect local homeomorphism |p_1(y)| is finite. Furthermore, Example 3.3, below, shows that there are surjective perfect local homeomorphisms between connected metric spaces that have points yo G Y at which \p~1(y)\ is not locally constant. Indeed, Example 3.4 shows, that there are DOMAINS OF PERFECT LOCAL HOMEOMORPHISMS 289

surjective perfect local homeomorphisms between metric spaces that have points yo G Y at which |p-1(y)| is not even locally bounded.

Exam ple 3.3. Let X = X\ U X 2 U X 3 C I 3 in the subspace topology where X\ is the unit circle X\ = {(:r,0,z) G 3 R : x2 + z2 = 1} andX 2 is the half­ open interval X 2 = {(0,y, 1) G R3 : 0 < y < |} andX 3 is the closed interval X 3 = {(0,y, —1) G R3 : 0 < y < 1}. LetY = Y\ U Y2 C R 3 where Yi is the unit circle Y\ = X\ and Y2 is the closed interval Y2 = X 3. Let p : X —> Y be the map defined by (p | X\) : X\ —> Y\ is the standard double cover of the circle to itself, (p | X 2) : X 2 Y2 is given by p(0,y,l) = (0 ,y ,-1), and (p | X 3) : X 3 -> Y2 is the identity. Then p is a local homeomorphism and is at most 2-to-l. So, by Lemma 3.1, p is perfect. However, |p-1(y)| is not locally constant at the point (0, —1) G Y. To the left of (0, —l), |p_1(y)| = 2 while starting at (0, —l) and proceeding to the right |p-1(y)| = 1. See Figure 3.3.

Figure 3.3

Example 3.4. Let

x = {(x ,i)e

U I I N(i,n)GM ^:- —

The domains of the maps are locally compact, locally path connected, semi-locally 1-connected metric spaces and in Example 3.3 the domain is even connected. Thus, adding hypotheses on the domain or range of a perfect local homeomorphism is unlikely to produce theorems as precise as those produced in [2] and [3] for proper local homeomorphisms. However, a couple of situations to examine are suggested by these examples. In neither, of 3.3 or 3.4 is the domain compact. Also, in Example 3.4 the domain is not connected. So there is the possibility that if the domain of p were required to be connected, a global bound on \p~1(y)\ could be obtained. Furthermore, one wonders what follows if one assumes that \p~1{y)\ is locally or globally constant on Y or if one assumes that \p~1(y)\ is locally or globally bounded on Y. The next results and examples deal with these questions. Examples 3.5 and 3.6 show the necessity of both the assumption that X be compact and p : X —► Y be a local homeomorphism in Theorem 3.7.

Exam ple 3.5. There are perfect maps p : X —► Y between compact metric spaces that are open and closed and yet are such that \p~1(y)\ is not globally bounded. Consider the sets X = {(0,l)}U{(l/n, 1) : n 6 N} andY = {(0,0) U {(1/n, 0) : n € N} both with the topology they inherit from the standard topology on M2. Both are compact. Define p : X —> Y by p(0,1) = (0,0), p(l, 1) = (1,1), p(l/k ,l) = (1/2,0) for 2 < k < 3, p(l/k, 1) = (1/3,0) for 4 < k < 6 , p(l/fc,1) = (1/4,0) for 7 < k < 10,___ The idea in this example is that the next n points in the first sequence, X, are mapped onto the n-th point (1,1/n) in the second sequence, Y. By Lemma 3.1, p is perfect. In fact, p is proper, closed, and open. As n —► +oo, |p-1(0,1/n) | —> +00 . However, p is not a local homeomorphism since there is no neighborhood of (0 , 1) that maps homeomorphically onto a neighborhood of (0 , 0 ).

Exam ple 3.6. This example describes a situation in which X and Y are connected and p a perfect local homeomorphism such that |p-1(y)| is unbounded on Y . Observe that in this situation \p~1{y)\ is locally bounded on Y. The domain X is a union of arcs giving the tree as is shown in Figure 3.6. Note that the arcs that are the ends of the branches of the tree are half open. The range Y is the union of circles as is shown in Figure 3.6. The topology on both X and Y is that inherited from the standard topology on the plane. The labelings of the arcs in X and in Y show the local homeomorphism p : X —> Y: all the arcs labeled with the integer n in the domain are mapped homeomorphically onto the single arc with label n in the range. The arrows on the arcs indicate the direction in which the identifications occurs. Observe that if you choose a sequence of points y2n+i £Y such that y2n+\ is in the interior of the (2n + l)-st arc in the range then for n > 0 , \p~1{yn)\ = n + 1 so that \p~1{y)\ is unbounded on Y yet each point z € Y has a neighborhood Vz about z such that |p_1(y)| is bounded on Vz. See Figure 3.6.

Theorem 3.7. Let p : X —> Y be a perfect local homeomorphism. If X is compact, then \p~1{y)\ is globally bounded on Y . Hence, \p~1{y)\ is locally bounded on Y . DOMAINS OF PERFECT LOCAL HOMEOMORPHISMS 291

Proof. Suppose that there is no global bound on \p~1(y)\. Then there is a sequence of points yn € p(X ) such that for each n € N, |p-1(yn)| > n. Let So = {yn : n € N}. Denote the closure of So by So- Then K = p~1(So) is a closed subset of the compact space X and is therefore compact. For each x 6 K, there is an open set Ux about x and an open set Vx about p(x) such that p \ Ux : Ux —► Vx is a surjective homeomorphism. But the collection {Ux : x € K} is an open cover of the compact set K. So there are finitely many, say UXl,... , UXM, that cover K. Now choose n > M + 1. Then |p-1(yn)| > n > M + 1. Say p~l (yn) = {^i,• ,zm, zm+i, • • • ,zn}- Since n > M + 1, some UXjl contains 2 or more of the Zfc’s. But for all k, p(zk) =yn and p | UXj, : UXj, —> VXj, is a surjective homeomorphism. So UXj/ contains at most one Zk- Contradiction. □

6 / I 2 4 6 ^8 / \ Jv /^ ) \

/ /> I I % 0 2 4 6 8 10 Jv /V / v / v, / K % 2 4 6 8 10 12 /^ /v, J s, t v Jv /

V V

Figure 3.6

One is tempted to try to prove that if p : X —> Y is a perfect local homeo­ morphism and the range Y is compact, then \p~1(y)\ is bounded on Y without any additional hypotheses on X. However, Example 3.4 indicates that additional hypotheses are required. In particular, requiring that X be connected seems promis­ ing. However, as Example 3.8 shows, this is not sufficient. E xam ple 3.8. Take the range of Example 3.6 and have the radii of the nth circle be ^ . Add the point to which these circles converge. LetX be the same set as in Example 3.6 and p : X —► Y the same map. Then X is connected, Y is compact and p is perfect but \p~1{y)\ is not bounded on Y. Note that p is not surjective. But, the domain can be modified to alleviate this problem. 292 MATHEW TIMM of Y U a half open axe. Extend p to X' via the identity on the copy of Y and the “obvious” map on the interval. See Figure 3.8 below.

2

c 2 V\ °> ' > 2 > 4 > 6 > 8 / \ / s, i v, J

V

Figure 3.8

Theorem 3.9. (a) Let X be Hausdorff and assume p : X —>Y is a perfect local homeomorphism. U \p~1{y)\ is locally constant on Y, then p is proper. (b) (Partial converse to 3.9 (a)). If p is proper, Y is connected, and Y is first countable then \p~1{y)\ is globally (and therefore locally) constant on Y. P roof. Suppose that |p-1(?/)| is locally constant onY. We show that this implies that p is closed. Let yo £ Y and suppose that there is an open set U C X such that p-1(yo) C U. Since p is perfect, |p-1(2/o)l is finite. Say p~x(yo) = - ,xn}. Since p is a local homeomorphism, there are neighborhoods Uj about Xj and a neighborhood Vo about yo such that p \ Uj : Uj —* Vo is a surjective homeomorphism and Uj C U. Since X is Hausdorff we may, without loss of generality, assume that Uj Pi Uk= 0 for j ^ k. Furthermore, since \p~1(y)\ is locally constant on Y and since [p—1 (2/0) I = n, there is a neighborhood V' about yo such that for every z £ V', \p~1(z)\ = n. Take V = Vq C\. V' Consider p-1(V). Clearly, P~1{yo) C p ~1(V). Furthermore, if z £ V,then \p~1(z)\ = n. But by construction, for each j, p \ Ujis onto Vo, so for each j , there is a Uj£ Uj such that p(uj)= z. Hence, p ~ l{z) = p ~ x{z) fl (U”=1C/j). Therefore, p ~ 1(V) C U. Hence, by Lemma 2.2, p is closed. Therefore, by Lemma 2.3, p is proper. The partial converse stated in the theorem follows immediately from [3]. □ E xam ple 3.10. Observe that even the partial converse to 3.9 (a) stated in 3.9 (b) is false if the domain is not Hausdorff. Let X = [0,1]x {0}U {(0,2)}. Put the standard topology on [0,1]x {0}. Use sets of the form {(0,2)}U{(a:, 0) x: 0< < a} for 0 < a < 1 to form a basis for X at (0,2). Note that X is not Hausdorff as there is no separation of (0,2) and (0,0) by disjoint open subsets of X. Let Y = [0,1]x {0} with the standard topology. Definep : X —> Y to be the projection p(xi,x2) = (#i,0). Then p is a perfect local homeomorphism between connected, compact spaces that fails to have \p~1(y)\ locally constant at (0,0) £ Y. This failure is due to the fact that the domain of p is not Hausdorff. DOMAINS OF PERFECT LOCAL HOMEOMORPHISMS 293

In fact, 3.9 (a) itself is false if the domain is not required to be Hausdorff. Let X' = X U (0,1] x {3} where the topology on the half-open interval (0,1] x {3} is the standard topology. Again, let Y be as above and take p : X' —* Y to be the projection p(xi,x2) = (x±, 0). Then p is a perfect local homeomorphism onto a connected, compact metric space with [p_1(?/)| = 2 for yall G F , However, p is not proper, since p~1(Y) is not compact.

E xam ple 3.11. The hypothesis that Y is connected is essential in the partial con­ verse of Theorem 3.9. Let Y be the disjoint union Y = where each Bm is an open interval with the standard topology. Let X = U+=1(U™= 1Amn) be another disjoint union of the open intervals Amn. Define the perfect local homeomorphism p : X —* Y to be any map that maps each Amn homeomorphically onto Bm.

Theorem 3.12. Let X be Hausdorff and p : X —> Y be an open map such that \p~1{y) I is locally constant and finite on Y. Then p is a proper local homeo­ morphism. In fact, if X and p{X) are first countable Hausdorff spaces then, p : X —* p(X) is a finite sheeted covering projection.

Proof. Let x G X. Let 2 = p(x). Then there is an n € N and a neighborhood V' about z such that for all y € V1, \p~1(y)\ = n. Let p~1(z) = {xi = x,x2,... ,^n}- Since X is Hausdorff there are neighborhoods Uj about Xj such that Uj f]U 'k= 0 , if j ^k. Furthermore, since p is open p{Uj) is an open neighborhood about z. Let V = V' f l p(U[) n ... n p(U'n) and let Uj = p_1(V) fl Uj. Note that for each j, Uj is an open set about Xj such that the restriction p \ Uj : Uj —> V is a surjective one-to-one open map from the open set Uj to the open set V. Hence, p is a local homeomorphism. It is perfect by Lemma 3.1 and so is proper by Theorem 3.9. Therefore, by [3] and [2],p(X) if is first countable, it follows that p : X —■> p(X) is a finite sheeted covering projection. □

The above results provide a detailed analysis of point-wise and local finiteness conditions on \p~1(y)\ for perfect local homeomorphisms. The examples given above, especially Examples 3.3 and 3.4, show that one cannot expect to obtain a global finiteness condition on the domain of a perfect local homeomorphism. Thus, the final result, Theorem 3.14, of this section, seems to be the best possible analog of both Theorem 1.1 and [3, 2.1] that one can obtain for perfect local home­ omorphisms. In addition, Theorem 3.14 provides the partial solution to Question 2.3 in Jungck and Timm [3], mentioned in the introduction. Note that Theorem 3.14 improves on the analog for proper maps since it does not require the domain and range of the map to be Hausdorff or first countable.

Lemma 3.13. Let X be a topological space. Then every component of X is open if and only if for every x £X there is a neighborhood Nx such that Nx intersects at most finitely many components of X.

P roof. Assume that every component of X is open. Let x G X. Let Cx denote the component of X. Then Cx is a neighborhood of x that intersects at most finitely many components of X. Now assume that for each x G X there is a neighborhood Nx about x that intersects at most finitely many components of X. Let Co be a component of X. Let x G Cq. Then there is a neighborhood Nx about x such that Nx intersects at most finitely many components of X. Say Nx intersects only 294 MATHEW TIMM components Co,. • • , Cn. Note Nx = (Nx fl Co) U ... U (Nx ft Cn), a disjoint union. Now x fi Ci U ... U Cn and since components are closed, Ci U ... U Cn is closed. Therefore NX\(C\ U ... U Cn) is a neighborhood of x contained entirely in Co- Therefore Co is open. □

Theorem 3.14. Let p : X —> Y be a perfect local homeomorphism. If Y is lo­ cally connected or for each y G Y at which Y is not locally connected there is a neighborhood V(y) about y such that p~1(V(y)) intersects at most finitely many components of X , then every component of X is open.

Proof. Y locally connected and p a local homeomorphism implies that X is also locally connected. So it follows immediately that every component of X is open. Now, assume that Y is not locally connected but for each y G Y at which Y is not locally connected there is a neighborhood V(y) about y such that p-1 (V(y)) intersects at most finitely many components of X. Let xo E X. If X is locally connected at xq then there is clearly a connected neighborhood Uq about x, and U0 intersects at most finitely many components of X. On the other hand, suppose that X is not locally connected at Xo. Then xq has a neighborhood Uq such that for any neighborhood U with xo G U C Uq, U is not connected. Since p is a local homeomorphism, there is a neighborhood N q about xo such that the restriction p | N q : N q —> p (Nq) is a homeomorphism onto a neighborhood of yo = p{xo). Let No = N q H Uq. Since p is a local homeomorphism it therefore follows that yo is point of Y at which Y is not locally connected. Hence, by assumption, there is a neighborhood Vo about yo = p(xo) such that p_1(Vo) intersects at most finitely components of X. So every point in X has a neighborhood that intersects at most finitely many components of X. Therefore, by Lemma 3.13, every component of X is open. □

There is an elementary example (Example 3.17) that shows that the there are perfect local homeomorphisms between metric spaces for which components of the domain need not be open even though the range is connected. The next example is a modification of Example 3.4 suggested by the referee.

E xam ple 3.15. This example show that that the converse of Theorem 3.14 is not true. Let X q be the set given by

{(x, 0,0) G R 3 : -1 < x < 1} U {(0,0,z) G R 3 : 0 < z < 1}

U {(0,j/, 1) G R3 : —1 < y < 0} U | ^ 0 ,- i , ^ G3 R: 0 < z < 1

Let X = Xo U ( U ^ ! {(x,n,0) G 3R : — - < x < 0V 0 < x < £}). Let Y = Xq. Consider X and Y as subspaces of R 3 with the standard topology. Let p : X —> Y be given by p(x,y,z) = (x,y,z), if y < 0, and p(x,y,z) = (x,0,z) if y > 0 . As with Example 3.4, p is a perfect surjective local homeomorphism, but \p~1(y)\ is not locally bounded at (0,0,0) G Y, which is not a point of local connectivity of Y. DOMAINS OF PERFECT LOCAL HOMEOMORPHISMS 295

An interesting problem is the following: P roblem 3.16. Are there any (somewhat natural) hypotheses that will yield both necessary and sufficient conditions for every component of the domain of a perfect local homeomorphism to a connected space to be open?

E xam ple 3.17. For k > 0, let

Xk = { ( * ’2 * ) : 0 “X ~ *} U :2 * _2*+2 < y < 2i* + 2^+2 }*

Let X = {(0,0)} U (U^f^Xfe), with the topology it inherits as a subspace of R 2. For n > 1, let Yn = {(a;, : 0

-----1

P V

Figure 3.17

4. Locally Proper Maps Theorems 3.9 and 3.12 are results that show that a perfect local homeomorphism, or more generally a perfect open map, that have additional “local compactness” conditions placed on them end up being proper. The last two observations of this paper deal with two obvious ways to impose “local compactness” conditions on the map. It was hoped that some such condition on the map p would give that every component of the domain of p is open and yet not imply that p is proper. The results of the first attempt to impose such a condition is Fact 4.1. This attempt also just gives another characterization of proper maps, at least when one assumes the range is Hausdorff. (Without the assumption that the range is Hausdorff, I am not sure what happens.) Example 4.2 deals with the standard definition of 296 MATHEW TIMM local compactness appearing in the literature and its relationship to Question 2.3 of Jungck and Timm [3].

Fact 4.1. If Y is Hausdorff and p : X —> Y is a map such that for each y E Y there is a compact set Ky, with y in the interior of Ky, such that p~1(Ky) is compact, then p is proper.

Proof. Let K be a compact subset of Y. Then K can be covered by finitely many of the sets Kyi, . . . , KVn that are mentioned in the statement of Fact 4.1. Since Y is Hausdorff, for each j = 1 ,... ,n, KI KyjC is a closed and, therefore compact, subset of Kyj so that p~l (Kyj fl K) is compact. Thus

n p~\K) = p - 1 MJ(K V> n K) K)) n ,k = l is a finite union of compact sets and is therefore compact. □

Example 4.2. Recall that a map p : X —► Y is said to be locally compact, or locally proper, if and only if for every x E X, there is a neighborhood U xabout x such that the restriction p | (cl(Ux)) : cl(Ux) —> p(c\(Ux)) is a proper map. Let Y be the subspace of R 2 given by

Y =

U {(x,0):iG [0,1]} U {(0,y) € K2 : y E [0,1]}.

Let X be the disjoint union of a copy of Y together with the subspace

Z = | U{(x,0): x E (0,1)}

Let p : X —> Y be defined by p(x, y) = (x, y) for all (x, y) E X. Then p is a perfect local homeomorphism from the metric space X onto the connected metric space Y that is a locally compact map, yet the component{(x, 0) : x E (0,1)} C Z C X is a component of X that is not open in X. So, even locally proper, perfect local homeomorphisms from a metric space onto a compact, connected metric space, need not be such that ever component of the domain is open. The reader should compare this example to Theorem 3.14 and observe that the failure of the theorem in this example is due to local connectivity problems in Z.

References 1. C.W. Ho, A Note on proper maps, Proc. Amer.Math. Soc. 51 No. 1 (1975), 237-241. 2. G.F. Jungck, Local Homeomorphisms, Dissertationes Mathematicae, CCIX, Polska Akademia Nauk, Instytut Matematyczny, Warszawa, 1983. 3. G.F. Jungck and M. Timm, Decompositions of domains of proper local homeo­ morphisms, Topology Proceedings, 17 (1992), 145-150. DOMAINS OF PERFECT LOCAL HOMEOMORPHISMS 297

4. L.F. McAuley and E.E. Robinson, Discrete open and closed mappings on gen­ eralized continua and Newman’s property, Can. J. Math. 36 No. 6 (1984), 1081-1112. Correction: Can. J. Math. 37 No. 4 (1985), 767-768. 5. R. Palais, When proper maps are closed, Proc. Amer. Math. Soc. 24 (1970), 835-836.

Mathew Timm Department of Mathematics Bradley University Peoria IL, 61625 USA [email protected]