Locally closed sets and submaximal spaces

Abstract

A topological space $X$ is called submaximal if every dense subset of $X$ is open. In this paper, we show that if $\beta X$, the Stone-\v{C}ech compactification of $X$, is a submaximal space, then $X$ is a compact space and hence $\beta X=X$. We also prove that if $\upsilon X$, the Hewitt realcompactification of $X$, is submaximal and first countable and $X$ is without isolated point, then $X$ is realcompact and hence $\upsilon X=X$. We observe that every submaximal Hausdorff space is pseudo-finite. It turns out that if $\upsilon X$ is a submaximal space, then $X$ is a pseudo-finite $\mu$-compact space. An example is given which shows that $X$ may be submaximal but $\upsilon X$ 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 ${\mathcal T_l}$. We study some topological properties between $(X,{\mathcal T})$ and $(X,{\mathcal T_l})$, such as we show that a) $(X,{\mathcal T_l})$ is discrete if and only if $(X,{\mathcal T})$ is a $T_D$-space; b) $(X,{\mathcal T})$ is a locally indiscrete space if and only if ${\mathcal T}={\mathcal T_l}$; c) $(X,{\mathcal T})$ is indiscrete space if and only if $(X,{\mathcal T_l})$ is connected. We see that, in locally indiscrete spaces, the concepts of $T_0$, $T_D$, $T_\frac{1}{2}$, $T_1$, submaximal and discrete coincide. Finally, we prove that every clopen subspace of an lc-compact space is lc-compact.