On topological spaces and topological groups with certain local countable networks
S. S. Gabriyelyan, J. Kakol

TL;DR
This paper investigates the local topological properties of spaces and groups with countable networks, characterizing metrizability and other properties, and applies these results to spaces of distributions, free topological groups, and locally convex spaces.
Contribution
It introduces new characterizations of spaces with countable $cp$- and $cn$-networks, linking them to metrizability, separability, and other structural properties, with applications to various functional spaces.
Findings
A Baire topological group is metrizable iff it has the strong Pytkeev property.
A topological group has a countable $cp$-network iff it is separable with such a network at the unit.
The space of distributions $D'(\Omega)$ has a countable $cp$-network, improving known tightness results.
Abstract
Being motivated by the study of the space of all continuous real-valued functions on a Tychonoff space with the compact-open topology, we introduced in [15] the concepts of a -network and a -network (at a point ) in . In the present paper we describe the topology of admitting a countable - or -network at a point . This description applies to provide new results about the strong Pytkeev property, already well recognized and applicable concept originally introduced by Tsaban and Zdomskyy [43]. We show that a Baire topological group is metrizable if and only if has the strong Pytkeev property. We prove also that a topological group has a countable -network if and only if is separable and has a countable -network at the unit. As an application we show, among the others, that the space of distributions over…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Banach Space Theory · Mathematical and Theoretical Analysis
