This talk is based on a joint work by T. A. Edwards, J. E. Joseph, M. H. Kwack and B. M. P. Nayar that apperared in the Journal of Advanced studies in Topology, Vol. 5 (4), 2014), 8 - 15. B
An adherence dominator on a topological space X is a function π from the collection of filterbases on X to the family of closed subsets of X satisfying A(Ω) ⊆ π(Ω) where A(Ω) is the adherence of Ω. The notations π(Ω) and A(Ω) are used for the values of the functions π and A and π(Ω) =⋂_Ω π F= ⋂_O π V, where O represents the open members of Ω. The π -adherence may be adherence,θ- adherence, u-adherence s-adherence,f- adherence δ-adherence etc., of a filterbase. Many of the recent theorems by the authors and others 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-semi-continuous relation β on X with π -strongly closed graph, the relation μ on X defined by μ = πβ has a maximal value with respect to set inclusion
