• Terrence A. Edwards University of the District of Columbia
• James E. Joseph Howard University
• Myung H. Kwack Howard University
• Bhamini M.P. Nayar Morgan Sate University

### 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  (\pi\Omega=\bigcap_\Omega\pi F=\bigcap_{ \mathcal O}\pi V\), where $$\mathcal O$$ represents the open members of  (\Omega\) . The $$\pi$$-adherence may be adherence,\  (\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].