SOME CRITERIA FOR Cp(X) TO BE AN LΣ(≤ ω)-SPACE

Given a cardinal. say that kappa is an L Sigma(< k)space (L Sigma(<= kappa)-space) if kappa has a countable network F with respect to a cover C of X by compact subspaces of weight strictly less than kappa ( less than or equal to kappa, respectively), i.e., given any C is an element of C, we have w(C) < kappa (w(C) <= kappa) and, for any U is an element of t( X) with C subset of U, there exists F. F such that C subset of F subset of U. These concepts were introduced and studied by Kubis, Okunev and Szeptycki. We show that if C-p(X) is a Lindel " of Sigma-space and |X| <= c, then C-p(X) is an L Sigma(<= omega)-space. This answers two questions of Kubis, Okunev and Szeptycki. We also prove that if X is a space and C-p(X) has the L Sigma(< omega)-property, then X is cosmic, i.e., nw(X) <= omega. This answers (in a stronger form) a question of Okunev published in Open Problems in Topology II.