On topological spaces and topological groups with certain local countable networks

Being motivated by the study of the space Cc(X)Cc(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∈Xx∈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)D′(Ω) of distributions over open Ω⊆RnΩ⊆Rn has a countable cp  -network, which essentially improves the well known fact stating that D′(Ω)D′(Ω) has countable tightness. We show that, if X   is an MKωMKω-space, then the free topological group F(X)F(X) and the free locally convex space L(X)L(X) have a countable cp-network. We prove that a topological vector space E is p  -normed (for some 0