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

Abstract

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

Keywords:

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

Key words:

• Compact support
• pseudocompact space
• intermediate ring
• pseudocompact support
• essential ideal
• z^◦-ideal
• socle
• C-type ring