Locally closed sets and submaximal spaces

Abstract

A subset $A$ of a topological space $X$ is called locally closed if it is open in $\ov{A}$; $X$ is called submaximal  if every subset of $X$ is locally closed. 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 observe that every submaximal Hausdorff space is a $ncd$-space (a space in which does not have a nonempty compact and dense in itself subset). It turns out that every dense in itself Hausdorff space is   pseudo-finite if and only if it is a $(cei,f)$-space (a space in which every compact subspace of $X$ with empty interior is finite). A new characterization for submaximal spaces is given. 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  $(X,{\mathcal T})$ is a locally indiscrete space  if and only if ${\mathcal T}={\mathcal T_l}$. Finally, we prove that every clopen subspace of an lc-compact space is  lc-compact.
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