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 language | English |
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Pages (from-to) | 10-29 |
Number of pages | 20 |
Journal | Topology and its Applications |
Volume | 227 |
DOIs | |
State | Published - 15 Aug 2017 |
Keywords
- Function space with compact-open topology
- Rectifiable space
- The strong Pytkeev property
- Topological group
- Topological loop
- Topological lop
ASJC Scopus subject areas
- Geometry and Topology