On topological spaces and topological groups with certain local countable networks

Being motivated by the study of the space C_c(X) of all continuous real-valued functions on a Tychonoff space X with the compact-open topology, we introduced in [16] the concepts of a cp-network and a cn-network (at a point x) in X. In the present paper we describe the topology of X admitting a countable cp- or cn-network at a point x∈X. This description applies to provide new results about the strong Pytkeev property, already well recognized and applicable concept originally introduced by Tsaban and Zdomskyy [44]. We show that a Baire topological group G is metrizable if and only if G has the strong Pytkeev property. We prove also that a topological group G has a countable cp-network if and only if G is separable and has a countable cp-network at the unit. As an application we show, among the others, that the space D^'(Ω) of distributions over open Ω⊆Rn has a countable cp-network, which essentially improves the well known fact stating that D^'(Ω) has countable tightness. We show that, if X is an MKω-space, then the free topological group F(X) and the free locally convex space L(X) have a countable cp-network. We prove that a topological vector space E is p-normed (for some 0<p≤1) if and only if E is Fréchet-Urysohn and admits a fundamental sequence of bounded sets.