On spaces defined by Pytkeev networks

Abstract

The notions of networks and k-networks for topological spaces have played an important role in general topology. Pytkeev networks, strict Pytkeev networks and cn-networks for topological spaces are defined by T. Banakh, and S. Gabriyelyan and J. Kakol, respectively. In this paper, we discuss the relationship among certain Pytkeev networks, strict Pytkeev networks, cn-networks and k-networks in a topological space, and detect their operational properties. It is proved that every point-countable Pytkeev network for a topological space is a quasi-k-network, and every topological space with a point-countable cn-network is a meta-Lindelöf D-space, which give an affirmative answer to the following problem [25, 29]: Is every Fréchet-Urysohn space with a pointcountable cs-network a meta-Lindelöf space? Some mapping theorems on the spaces with certain Pytkeev networks are established and it is showed that (strict) Pytkeev networks are preserved by closed mappings and finite-to-one pseudo-open mappings, and cn-networks are preserved by pseudo-open mappings, in particular, spaces with a point-countable Pytkeev network are preserved by closed mappings.

Some Results on Pixley–Roy Hyperspaces

“…(1) P is a network [17] for X, if for any neighborhood U of a point x ∈ X, there exists a set P ∈ P such that x ∈ P ⊂ U (2) P is a k-network [18] for X, if whenever K is a compact subset of an open set U in X, there exists a finite subfamily F ⊂ P such that K ⊂ ⋃ F ⊂ U (3) P is a cn-network [19] for X, if for any neighborhood U of a point x ∈ X, the set ⋃ P ∈ P: x ∈ { P ⊂ U} is a neighborhood of x (4) P is a ck-network [19] for X, if for any neighborhood U of a point x ∈ X, there is a neighborhood U x of x such that for each compact subset K ⊂ U x , there exists a finite subfamily F ⊂ P satisfying x ∈ ∩ F and K ⊂ ⋃ F ⊂ U (5) P is a cp-network [19] for X, if for each x ∈ X if either x is an isolated point of X and x { } ∈ P, or for each subset A of X with x ∈ A∖A and for each neighborhood U of x there exists a set P ∈ P such that P ∩ A is infinite and x ∈ P ⊂ U (6) P is a cs * -network [15] in X if for each x ∈ X, each neighborhood U of x and any sequence (x n ) ⊂ X converging to x there is P ∈ P such that P ⊂ U and P contains infinitely many members of the sequence (x n ) (7) P is a Pytkeev network [20][21][22] (resp., strict Pytkeev network) [15] for X, if P is a network for X, and for each neighborhood U of a point x in X and each subset A of X accumulating at x, there exists a set P ∈ P such that P ∩ A is infinite and P ⊂ U (resp.,…”

The strong Pytkeev property and strong countable completeness in (strongly) topological gyrogroups

“…The strong Pytkeev property is usually studied combining the other spaces, such as topological groups, topological vector spaces, etc., see [11,20,21,28,32]. In this paper, we mainly research the strong Pytkeev property in topological gyrogroups.…”

The strong Pytkeev property and strong countable completeness in (strongly) topological gyrogroups

“…The strong Pytkeev property is usually studied combining the other spaces, such as topological groups, topological vector spaces, etc., see [11,20,21,28,32]. In this paper, we mainly research the strong Pytkeev property in topological gyrogroups.…”