The strong Pytkeev property in topological spaces

Taras Banakh, Arkady Leiderman

Research output: Contribution to journalArticlepeer-review

Abstract

A topological space X has the strong Pytkeev property at a point x∈X if there exists a countable family N of subsets of X such that for each neighborhood Ox⊂X and subset A⊂X accumulating at x, there is a set N∈N such that N⊂Ox and N∩A is infinite. We prove that for any ℵ0-space X and any space Y with the strong Pytkeev property at a point y∈Y the function space Ck(X,Y) has the strong Pytkeev property at the constant function X→{y}⊂Y. If the space Y is rectifiable, then the function space Ck(X,Y) is rectifiable and has the strong Pytkeev property at each point. We also prove that for any pointed spaces (Xn,⁎n), n∈ω, with the strong Pytkeev property their Tychonoff product ∏n∈ωXn and their small box-product ⊡n∈ωXn both have the strong Pytkeev property at the distinguished point (⁎n)n∈ω. We prove that a sequential rectifiable space X has the strong Pytkeev property if and only if X is metrizable or contains a clopen submetrizable kω-subspace. A locally precompact topological group is metrizable if and only if it contains a dense subgroup with the strong Pytkeev property.

Original languageAmerican English
Pages (from-to)10-29
Number of pages20
JournalTopology and its Applications
Volume227
DOIs
StatePublished - 15 Aug 2017

Keywords

  • Function space with compact-open topology
  • Rectifiable space
  • The strong Pytkeev property
  • Topological group
  • Topological loop
  • Topological lop

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

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