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

Original · EN

For any ideal P of closed sets in X, let Cₚ(X) be the family of those functions in C(X) whose support lie on P. Further let Cᵖ∞(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 υCₚX stand for the set of all those points p in βX at which the stone extension f* for each f in Cₚ(X) is real valued. We show that each realcompact space lying between X and βX is of the form υCₚX if and only if X is pseudocompact. We find out conditions under which an arbitrary product of spaces of the form locally-P/ almost locally-P, becomes a space of the same form. We further show that Cₚ(X) is a free ideal (essential ideal) of C(X) if and only if Cᵖ∞(X) is a free ideal (respectively essential ideal) of C*(X)+Cᵖ∞(X) when and only when X is locally-P (almost locally-P). We address the problem, when does Cₚ(X)/Cᵖ∞(X) become identical to the socle of the ring C(X). Finally we observe that the ideals of the form Cₚ(X) of C(X) are no other than the z∘-ideals of C(X).