On Countably Uniform Closed-Spaces

Let C_c(X)={f: f(X) is countable}, Z_c(X)={Z(f): f∈C_c(X)} and S⊆ X is Z_c-embedded if each zero-set of S is the restriction a zero-set of X. We call X is a countably uniform closed space, briefly, a CUC-space, if C_c(X) is closed under uniform convergence. We investigate some properties of CUC -spaces. It is observed that in zero-dimensional CUC-space, eachLindelof subspae is Z_c-embedded. Finally, we show that if each Lindelof subspace of CUC-space is c -completely separated from each disjoint zero-set, then it is C_c-embedded.