Epireflective subcategories of TOP, T $$_2$$ 2 UNIF, UNIF, closed under epimorphic images, or being algebraic

Abstract

The epireflective subcategories of \(\mathbf{Top}\), that are closed under epimorphic (or bimorphic) images, are \(\{ X \mid |X| \le 1 \} \), \(\{ X \mid X\) is indiscrete\(\} \) and \(\mathbf{Top}\). The epireflective subcategories of \(\mathbf{T_2Unif}\), closed under epimorphic images, are: \(\{ X \mid |X| \le 1 \} \), \(\{ X \mid X\) is compact \(T_2 \} \), \(\{ X \mid \) covering character of X is \( \le \lambda _0 \} \) (where \(\lambda _0\) is an infinite cardinal), and \(\mathbf{T_2Unif}\). The epireflective subcategories of \(\mathbf{Unif}\), closed under epimorphic (or bimorphic) images, are: \(\{ X \mid |X| \le 1 \} \), \(\{ X \mid X\) is indiscrete\(\} \), \(\{ X \mid \) covering character of X is \( \le \lambda _0 \} \) (where \(\lambda _0\) is an infinite cardinal), and \(\mathbf{Unif}\). The epireflective subcategories of \(\mathbf{Top}\), that are algebraic categories, are \(\{ X \mid |X| \le 1 \} \), and \(\{ X \mid X\) is indiscrete\(\} \). The subcategories of \(\mathbf{Unif}\), closed under products and closed subspaces and being varietal, are \(\{ X \mid |X| \le 1 \} \), \(\{ X \mid X\) is indiscrete\(\} \), \(\{ X \mid X\) is compact \(T_2 \} \). The subcategories of \(\mathbf{Unif}\), closed under products and closed subspaces and being algebraic, are \(\{ X \mid X\) is indiscrete\( \} \), and all epireflective subcategories of \(\{ X \mid X\) is compact \(T_2 \} \). Also we give a sharpened form of a theorem of Kannan-Soundararajan about classes of \(T_3\) spaces, closed for products, closed subspaces and surjective images.