The strong Pytkeev property in topological spaces

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 ℵ₀-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∈ωX_n and their small box-product ⊡_n∈ωX_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.