On Locally Closed Sets and some Related Topological Spaces

Authors

  • Rostam Mohamadian Shahid Chamran University of Ahvaz
  • A.R. Aliabad
  • S. Taram

Abstract

A subset $A$ of a topological space $X$ is said to be locally closed if it can be represented as the intersection of an open set and a closed set in $X$. The family of all locally closed subsets forms a basis for a topology, denoted by $\ell(\CT)$. We study topological relationships between $(X, \CT)$ and $(X, \ell(\CT))$. We examine the behavior of locally closed sets under various separation axioms and in special classes of spaces such as locally indiscrete spaces, door spaces, submaximal spaces, Alexandroff spaces, etc. We investigate the behavior of ``$\ell$" as a map. For example, we show that the map ``$\ell$" distributes over the free union of topologies and also over the product of topologies. In this regard, we will show that $\ell^2(\CT)=\ell^3(\CT)$ for any topology $\CT$ on $X$.

Published

2026-09-16

Issue

Vol. 21 (2027)

Section

No. 1

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