Supports of continuous functions

Mark Mandelker
1971 Transactions of the American Mathematical Society  
Gillman and Jerison have shown that when A" is a realcompact space, every function in C(X) that belongs to all the free maximal ideals has compact support. A space with the latter property will be called fi-compact. In this paper we give several characterizations of /¿-compact spaces and also introduce and study a related class of spaces, the ^-compact spaces ; these are spaces X with the property that every function in C(X) with pseudocompact support has compact support. It is shown that every
more » ... realcompact space is ^-compact and every i/i-compact space is /¿-compact. A family & of subsets of a space X is said to be stable if every function in C(X) is bounded on some member of #". We show that a completely regular Hausdorff space is realcompact if and only if every stable family of closed subsets with the finite intersection property has nonempty intersection. We adopt this condition as the definition of realcompactness for arbitrary (not necessarily completely regular Hausdorff) spaces, determine some of the properties of these realcompact spaces, and construct a realcompactification of an arbitrary space. 1. Introduction. The support of a real continuous function / on a topological space A" is the closure of the set of points in Afat which/does not vanish. Gillman and Jerison have shown that when A'is a realcompact space, the functions in C(X) with compact support are precisely the functions which belong to every free maximal ideal in C(X). This result, and other general background material, may be found in our basic reference [GJ]. A space with the property of the Gillman-Jerison result will be said to be pcompact. Other writers have shown that discrete spaces (Kaplansky [Ki, Theorem 28]), P-spaces (Kohls [K2, Theorem 3.9]), and spaces that admit complete uniform structures (Robinson [Ri]) are /x-compact. Examples given in [GJ] show that not every space is /^-compact, and not every /x-compact space is realcompact. In this paper we show that a third class of spaces may be interpolated between the realcompact and /x-compact. This class, of ip-compact spaces, consists of those spaces X for which every function in C(X) with pseudocompact support has compact support. Examples will be given to show that the three classes of spaces are distinct. Every P-space (hence every discrete space) and every space that admits a
doi:10.1090/s0002-9947-1971-0275367-4 fatcat:oqrpr4ck35emna6ge4sqsw7kiu