2022/05/15 by R. Mohamadian, Mohamadian, Rostam
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #General Topology (math.GN)
paper · pdf · doi:10.48550/arxiv.2205.07191
openalex publication_date 2022/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A topological space X is called submaximal if every dense subset of X is open. In this paper, we show that if βX, the Stone-Čech compactification of X, is a submaximal space, then X is a compact space and hence βX=X. We also prove that if υ X, the Hewitt realcompactification of X, is submaximal and first countable and X is without isolated point, then X is realcompact and hence υ X=X. We observe that every submaximal Hausdorff space is pseudo-finite. It turns out that if υ X is a submaximal space, then X is a pseudo-finite μ-compact space. An example is given which shows that X may be submaximal but υ X may not be submaximal. Given a topological space (X,\mathcal T), the collection of all locally closed subsets of X forms a base for a topology on X which is denotes by \mathcal Tl. We study some topological properties between (X,\mathcal T) and (X,\mathcal Tl), such as we show that a) (X,\mathcal Tl) is discrete if and only if (X,\mathcal T) is a TD-space; b) (X,\mathcal T) is a locally indiscrete space if and only if \mathcal T=\mathcal Tl; c) (X,\mathcal T) is indiscrete space if and only if (X,\mathcal Tl) is connected. We see that, in locally indiscrete spaces, the concepts of T0, TD, T_(1)/(2), T1, submaximal and discrete coincide. Finally, we prove that every clopen subspace of an lc-compact space is lc-compact.