Title: On -compactlike spaces and reflective subcategories
Abstract: We introduce the new topology on a topological space generated by the -sets. For an extensive subcategory of a category A of Hausdorff spaces and continuous maps, we consider the subcategory of A determined by those members of A which are -closed in their -reflection spaces. It is shown that is also an extensive subcategory of A for every infinite cardinal number if A is hereditary. It is also shown that a completely regular space is -closed in its Stone-Čech compactification iff it is -compact, that a zero-dimensional space is -closed in its maximal zero-dimensional compactification iff every maximal open closed filter with the -intersection property is fixed, that a Hausdorff uniform space is -closed in its completion iff every Cauchy filter with the -intersection property is convergent, and that a Hausdorff space is -closed in its Katětov extension iff every maximal open filter with the -intersection property is convergent.