Abstract
Abstract so-metrizable spaces are a class of important generalized metric spaces between metric spaces and s n sn -metrizable spaces where a space is called an so-metrizable space if it has a σ \sigma -locally finite so-network. As the further work that attaches to the celebrated Alexandrov conjecture, it is interesting to characterize so-metrizable spaces by images of metric spaces. This paper gives such characterizations for so-metrizable spaces. More precisely, this paper introduces so-open mappings and uses the “Pomomarev’s method” to prove that a space X X is an so-metrizable space if and only if it is an so-open, compact-covering, σ \sigma -image of a metric space, if and only if it is an so-open, σ \sigma -image of a metric space. In addition, it is shown that so-open mapping is a simplified form of s n sn -open mapping (resp. 2-sequence-covering mapping if the domain is metrizable). Results of this paper give some new characterizations of so-metrizable spaces and establish some equivalent relations among so-open mapping, s n sn -open mapping and 2-sequence-covering mapping, which further enrich and deepen generalized metric space theory.
Highlights
As a class of important generalized metric spaces between sn-metrizable spaces and metric spaces, sometrizable spaces were introduced by Ge in [1]
A space is an so-metrizable space [1] if it has a σlocally finite so-network
Having gained some enlightenments from previous work of Ponomarev [9] and Lin and Yan [8], we introduce so-open mappings and use the “Pomomarev’s method” to prove that a space X is an so-metrizable space if and only if X is an so-open, compact-covering, σ-image of a metric space, if and only if X is an so-open, σ-image of a metric space
Summary
As a class of important generalized metric spaces between sn-metrizable spaces and metric spaces, sometrizable spaces were introduced by Ge in [1]. Lin and Yan introduced σ-mapping and used “Ponomarev’s method” to characterize g-metrizable spaces by compact-covering, quotient, compact, σ-images of metric spaces [8]. “Ponomarev’s method” was established by Ponomarev in order to characterize first countable spaces by open images of metric spaces [9] The key of this method is to construct a metric space M and a mapping f : M ⟶ X such that f (M) = X, where X is a given generalized metric space. Results of this paper give some new characterizations of sometrizable spaces and establish some equivalent relations among so-open mapping, sn-open mapping and 2-sequence-covering mapping, which further enrich and deepen generalized metric space theory.
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