On a type of compactness via grills

Abstract

In this paper, we introduce and study the idea of a new type ofcompactness, defined in terms of a grill $\cal G$ in atopological space $X$. Calling it $\cal G$-compactness, weinvestigate its relation with compactness, among other things.Analogues of Alexender's subbase theorem and Tychonoff producttheorem are also obtained for $\cal G$-compactness. Finally,we exhibit a new method, in terms of the deliberations here, forconstruction of the well known one-point compactification of a$T_2$, locally compact and noncompact topological space.