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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 everydoi:10.2307/1995598 fatcat:whgixjpn5vhlhd56yuqii3ma7i