Proper mappings and dimension

Abstract

In this note it is shown that if W is the long line and f is a proper mapping of WX [0, 1 ] into Y, then dim Y22. This answers a question raised by Isbell. The main result of this note is that if W is the long line and f is a proper mapping of WX [0, 1) into Y, then dim Y_2. This answers a question posed by Isbell [2, pp. 119-120]. The main result will follow after a few preliminary results. LEMMA. Let X= [0, l]X [0, 1) with CO= {O} X [0, 1) and C1= {1 X [0, 1). If D is a closed set which separates CO and C1 in X, then D has a noncompact component. PROOF. Suppose that X-D = UU V is a separation with CoC U and C1C V. There is a continuousf: UkJD[0, 1] which is 0 on Co and 1 on D. There is a continuous function g: VUD-> [1, 2] which is 1 on D and 2 on C1. Thus there is a continuous function h:X ->[0, 2] which is 0 on C0, 1 on D, and 2 on C1. Accordingly there is a homeomorphism i:X-*X taking Co to Co, C1 to C1, and D to a set E whose closure in R2 is disjoint from Zo and Cl. Then E separates the two sides of [0, 1 ] X [0, 1 ] and thus one of its components K intersects both the top and bottom of the square. Now K contains a component of E and it cannot contain a compact component of E. Therefore E and thus D have noncompact components. PROPOSITION. Let X, Co, and C1 be as in the Lemma. If f(X) = Y is a proper mapping withf(Co) Cf(Ci) = 0, then dim Y_ 2. PROOF. We note that Y is separable metric by [6]. Thus the various dimension functions coincide as proved in [1]. We will show that Ind Y_ 2 by showing that f(C0) and f(C1) cannot be separated in Y by a closed 0-dimensional set. Suppose that E is closed and separates f(Co) andf(C1) in Y. Thenf-I(E) is closed in X and separates Co and C1. By the Lemma, f-'(E) has a noncompact component D. Since f is proper and D is closed, f cannot be constant on D. Therefore f(D) is connected and not a point and thus dim f(D) > 1 and dim E> 1. This proves the Proposition. Received by the editors July 22, 1970. AMS 1970 subject classifications. Primary 54F45; Secondary 54CGb.