On countably uniform closed-spaces

Abstract

Let X be a topological space and C_c(X) be the functionally countable subalgbera of C(X). We call X to be a countably uniform closed-space, briefly, a CU C-space, if C_c(X) is closed under uniform convergence. We investigate that countably uniform closedness need not closed under finite intersection and infinite product. It is shown that if X is a countable union of quasi-components, then X is a CU C-space. We characterize C_c-embedding and also -embedding in CU C-spaces. A subset S of X is called Z_c-embedded, if each Z ∈ Z_c(S) is the restriction of a zero-set of Z_c(X). It is observed that in a zero-dimensional CU C-space, each Lindelöf subspae is Z_c-embedded. Moreover, it is shown that in CU C-spaces, each Lindelöf subspace is C_c-embedded if and only if it is c-completely separated from each zero-set, which is disjoint from it. Also in latter spaces, it is observed that for each S ⊆ X, C_c-embedding, -embedding and Z_c-embedding coincide, when S belongs to Z_c(X) or it is a c-pseudocompact space. Finally, when X is both a CU C-space and a CP-space, then each Z_c-embedded subspace is C_c-embedded (-embedded) in X.