$SI_2$-quasicontinuous spaces

TL;DR

This paper introduces $SI_2$-quasicontinuous spaces, generalizing previous concepts, and characterizes their properties through $ ext{GD}$-convergence and hypercontinuity, advancing the understanding of topological space structures.

Contribution

It defines $SI_2$-quasicontinuous spaces and characterizes them via weakly irreducible topology and $ ext{GD}$-convergence, linking these concepts to hypercontinuity.

Findings

A space is $SI_2$-quasicontinuous iff its weakly irreducible topology is hypercontinuous.

$T_0$ space $X$ is $SI_2$-quasicontinuous iff $ ext{GD}$-convergence is topological.

Characterizations of $SI_2$-quasicontinuity in terms of topology and convergence.

Abstract

In this paper, as a common generalization of $SI_{2}$-continuous spaces and $s_{2}$-quasicontinuous posets, we introduce the concepts of $SI_{2}$-quasicontinuous spaces and $\mathcal{GD}$-convergence of nets for arbitrary topological spaces by the cuts. Some characterizations of $SI_{2}$-quasicontinuity of spaces are given. The main results are: (1) a space is $SI_{2}$-quasicontinuous if and only if its weakly irreducible topology is hypercontinuous under inclusion order; (2) A $T_{0}$ space $X$ is $SI_{2}$-quasicontinuous if and only if the $\mathcal{GD}$-convergence in $X$ is topological.