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Domains of Perfect Local Homeomorphisms 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.
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