Some new results on functions in C(X) having their support on ideals of closed sets

Abstract

For any ideal P of closed sets in X, let C_P(X) be the family of those functions in C(X) whose support lie on P. Further let C ^P_∞ (X) contain precisely those functions f in C(X) for which for each ϵ > 0, {x ∈ X : |f(x)| ≥ ϵ} is a member of P. Let ^υC_PX stand for the set of all those points p in βX at which the stone extension f* for each f in C_P(X) is real valued. We show that each realcompact space lying between X and βX is of the form ^υC_PX if and only if X is pseudocompact. We find out conditions under which an arbitrary product of spaces of the form locally-P or almost locally-P, becomes a space of the same form. We further show that C_P(X) is a free ideal (essential ideal) of C(X) if and only if C ^P_∞ (X) is a free ideal (essential ideal) of C* (X) + C ^P_∞ (X) when and only when X is locally-P (almost locally-P). We address the problem, when does C_P(X) or C ^P_∞ (X) become identical to the socle of the ring C(X). The results obtained turn out to imply a special version of the fact obtained by Azarpanah corresponding to the choice P ≡ the ideal of compact sets in X. Finally we observe that the ideals of the form C_P(X) of C(X) are no other than the z^◦ -ideals of C(X).

Mathematics Subject Classification (2010): Primary 54C40; Secondary 46E25.

Keywords: Compact support, pseudocompact space, intermediate ring, pseudocompact support, essential ideal, z^◦-ideal, socle, C-type ring