On topological spaces and topological groups with certain local countable networks

S. Gabriyelyan, J. Kakol

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


Being motivated by the study of the space 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∈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.

Original languageEnglish
Pages (from-to)59-73
Number of pages15
JournalTopology and its Applications
StatePublished - 1 Aug 2015


  • (Free) locally convex space
  • Baire space
  • Cn-network
  • Free (abelian) topological group
  • Function space
  • Small base
  • The strong Pytkeev property
  • Topological group

ASJC Scopus subject areas

  • Geometry and Topology


Dive into the research topics of 'On topological spaces and topological groups with certain local countable networks'. Together they form a unique fingerprint.

Cite this