# Compactness Via Adherence Dominators

## Main Article Content

## Abstract

An adherence dominator on a topological space \(X\) is a function \(\pi\) from the collection of filterbases on \(X\) to the family of closed subsets of \(X\) satisfying \(\mathcal A(\Omega) \subset \pi(\Omega)\) where \(\mathcal A(\Omega)\) is the adherence of \(\Omega\) [12]. The notations \(\pi\Omega\) and \(\mathcal A\Omega\) are used for the values of the functions \(\pi\) and \(\mathcal A\) and Â (\pi\Omega=\bigcap_\Omega\pi F=\bigcap_{ \mathcal O}\pi V\), where \(\mathcal O\) represents the open members of Â (\Omega\) . The \(\pi\)-adherence may be adherence,\ Â (\theta\)-adherence [21], \(u\)-adherence [5,6,9], \(s\)-adherence [8,11], \(f\)-adherence [7,12] \(\delta\) adherence [19], etc., of a filterbase. Many of the theorems in [2,3] and [16] on Hausdorff-closed, Urysohn-closed, and regular-closed spaces are subsumed in this paper. It is also shown that a space \(X\) is compact if and only if for each upper-semicontinuous \(\lambda\) on \(X\) with \(\pi\) strongly closed graph, the relation \(\mu\) on \(X\) defined by \(\mu=\pi\lambda\) has a maximal value with respect to set inclusion, genalizing results in [4,5].

## Article Details

*Journal of Advanced Studies in Topology*,

*5*(4), 8–15. Retrieved from http://m-sciences.com/index.php/jast/article/view/163

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