## 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 O_{x}⊂X and subset A⊂X accumulating at x, there is a set N∈N such that N⊂O_{x} and N∩A is infinite. We prove that for any ℵ_{0}-space X and any space Y with the strong Pytkeev property at a point y∈Y the function space C_{k}(X,Y) has the strong Pytkeev property at the constant function X→{y}⊂Y. If the space Y is rectifiable, then the function space C_{k}(X,Y) is rectifiable and has the strong Pytkeev property at each point. We also prove that for any pointed spaces (X_{n},⁎_{n}), n∈ω, with the strong Pytkeev property their Tychonoff product ∏_{n∈ω}X_{n} and their small box-product ⊡_{n∈ω}X_{n} 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 k_{ω}-subspace. A locally precompact topological group is metrizable if and only if it contains a dense subgroup with the strong Pytkeev property.

Original language | American 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

## All Science Journal Classification (ASJC) codes

- Geometry and Topology