Strict Pytkeev networks with sensors and their applications in topological groups

Abstract

Abstract Based on the notions of T. Banakh s strict Pytkeev networks and A.V. Arhangel skii s sensor families, strict Pytkeev networks with sensors are introduced in this paper. A family P of subsets of a topological space X is called a strict Pytkeev network with sensors (abbr. an sp-network) if, for each x ∈ U ∩ A ‾ with U open and A subset in X, there is a set P ∈ P such that x ∈ P ⊂ U and x ∈ P ∩ A ‾ . In present paper, we discuss certain relationship and operations among spaces defined by special Pytkeev networks, study spaces with a point-countable sp-network and spaces with a σ-closure-preserving sp-network, and detect some applications of sp-networks in topological groups. The following results are obtained: (1) Every sp-network is preserved by a continuous pseudo-open mapping. (2) Every k-space with a point-countable sp-network coincides with a continuous pseudo-open s-image of a metric space. (3) Every regular feebly compact space with a point-countable sp-network has a point-countable base. (4) A regular space has a countable sp-network if and only if it is separable and has a point-countable sp-network. (5) A topological space is stratifiable if and only if it is a regular space with a σ-closure-preserving sp-network. (6) A regular space with a σ-locally finite sp-network has a σ-discrete sp-network. (7) A topological group is metrizable if it has countable sp-character. (8) There is a non-Frechet-Urysohn sequential topological group with a countable strict Pytkeev network, which give a negative answer to a question posed by A.V. Arhangel skii [1] .