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• W2015444748 abstract "In this paper we give some characterizations of functionally compact Hausdorff spaces in terms of multifunctions. Also, some applications are presented. Introduction. Hausdorff functionally compact spaces have been introduced in 1969 by Dickman and Zame [2]. Some characterizations of such spaces, in terms of covers, have been given by Goss and Viglino [5] and by Lim and Tan [8]. Other characterizations, in terms of nets, have been found by Herrington [4]. The main purpose of this note is to establish a characterization of Hausdorff functionally compact spaces in terms of multifunctions. From our main result, we then obtain some generalizations of a theorem proved by Herrington and Long in [6]. 1. Preliminaries. In the sequel, we will follow the notation of [3 and 9]. We recall only some definitions. DEFINITION 1.1. Let X be a topological space. A filterbase in X is a family 1 of subsets of X with the property that for every F1, F2 E . there is F3 E . such that F3 C F1 n F2. It is allowed that 0 E F. The adherence of the filterbase 1 is the set A = nF F. We say that 1 converges to A if for every open set V C X, with A C V, there exists F E . such that F C V. DEFINITION 1. 2. Let X, Y be two topological spaces. A multifunction 41 from X into Y (briefly, 4D: X -* 2Y) is a function from X into the family of all subsets of Y. For every A C X, and B C Y, we put 40(A) = UXeA ?(x) and 40-1(B) = {x E x: ?(x) n B f 0}. We say that 4D is open (resp. closed) if 40(A) is open (resp. closed) in Y for every open (resp. closed) set A C X. We say that 4D is upper (resp. lower) semicontinuous if 40-l(B) is closed (resp. open) in X for every closed (resp. open) set B C Y. In particular, observe that 4D is upper semicontinuous if and only if for every x E X and every open set V C Y, with 40(x) C V, there exists U E Bz (B9 denotes the family of all open neighborhoods of x) such that 40(U) C V. We say that 4D is continuous if it is upper and lower semicontinuous. The graph of 4D is the set Gr(4D) = {(x, y) E X x Y: y E 40(x)}. We say that 41 has a closed graph if Gr(4D) is closed in X x Y. It is well known that the following characterization holds: PROPOSITION 1.3. Let X and Y be two topological spaces. Then, a multifunction 4D: X -* 2Y has a closed graph if and only if 4?(x) = nuEB ?(U) for all x E X. DEFINITION 1.4. Let X be a topological space and A C X. We put cl10(A) = x E X: VU E B9UnA : 0}. Received by the editors June 3, 1985 and, in revised form, October 10, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 54D30, 54C60. (@)1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page" @default.
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• W2015444748 title "A note on functionally compact spaces" @default.
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• W2015444748 doi "https://doi.org/10.1090/s0002-9939-1986-0857951-6" @default.
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