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 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.