The strong Pytkeev property in topological spaces

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.