Generalized Metric Spaces and Mappings

Abstract

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On spaces defined by Pytkeev networks

“…Next, suppose a regular k-space X with a locally countable k-network. By [28,Corollary 2.8.11], X is a topological sum of sequential spaces with a countable cs * -network. By Theorem 3.5, X is a topological sum of spaces with a countable Pytkeev network, i.e., X is a topological sum of P 0 -spaces.…”

“…As we know, metrization theory is the core in the study of general topology, and the theory of generalized metric spaces is an important generalization of this theory [19,28]. The ideas and problems of generalized metric spaces, in particular, the general metrization problem, greatly influenced all domains of set-theoretic topology [22].…”

Feathered gyrogroups and gyrogroups with countable pseudocharacter

“…Let N be the set of all positive integers and ω the first infinite ordinal. The readers may consult [1,5,11] for notations and terminologies not explicitly given here. Next we recall some definitions and facts.…”

Sequentially compact spaces with a point-countable k-network

“…(2) P is a k-network [15] for X if whenever K is a compact subset of an open set U in X, there exists an F ∈ P <ω such that K ⊂ ∪F ⊂ U . (3) P is a p-meta-base [5,11] for X if for any two distinct points x, y ∈ X, there exists an F ∈ P <ω such that x ∈ (∪F) • ⊂ ∪F ⊂ X \ {y}. (4) P is a p-k-network [9,11] for X if whenever K is compact and y ∈ X \ K, there exists an F ∈ P <ω such that K ⊂ ∪F ⊂ X \ {y}.…”

“…For the sake of brevity, the second author in this paper inspired by the concept of "p-base" in [4] first used the terminology "p-meta-base" and "p-k-network" in his monograph [11]. (2) It is easy to see that every base of a space is a p-meta-base and k-network, every p-meta-base or k-network of a space is a p-k-network, and every k-network of a space is a network.…”