$\mathfrak{E}$ of topological spaces II

Abstract

In this paper, we obtain analogues, in the situation of$\mathfrak{E}$-extensions, of Magill's theorem on lattices of compactifications. We define an epireflective subcategory of the categoryT 2 of all Hausdorff spaces to be admissive (respectively finitely admissive) if for any$\mathfrak{E}$-regular spaceX, every Hausdorff quotient of$\beta _\mathfrak{E} X$ which is Urysohn on$\beta _\mathfrak{E} X - X$ (respectively which is finitary on$\beta _\mathfrak{E} X - X$) and which is identity onX, has$\mathfrak{E}$. We notice that there are many proper epireflective subcategories ofT 2 containing all compact spaces and which are admissive; there are many such which are not admissive but finitely admissive. We prove that when$\mathfrak{E}$ is a finitely admissive epireflective subcategory ofT 2, then the lattices of finitary$\mathfrak{E}$-extensions of two spacesX andY are isomorphic if and only if$\beta _\mathfrak{E} X - X$ and$\beta _\mathfrak{E} Y - Y$ are homeomorphic. Further if$\mathfrak{E}$ is admissive, then the lattices of Urysohn$\mathfrak{E}$-extensions ofX andY are isomorphic if and only if$\beta _\mathfrak{E} X - X$ and$\beta _\mathfrak{E} Y - Y$ are homeomorphic.