When the Family of Functions Vanishing at Infinity Is an Ideal of $C(X)$

Abstract

We prove that C-infinity(X) is an ideal in C(X) if and only if every open locally compact subset of X is bounded. In particular, if X is a locally compact Hausdorff space, C, (X) is an ideal of C-infinity(X) if and only if X is a pseudocompact space. It is shown that the existence of some special functions in C-infinity(X) causes C-infinity(X) not to be an ideal of C(X). Finally we will characterize the spaces X for which C-infinity(X) and C-K(X), or C-psi(X), coincide.