Epireflective subcategories of Top, $T_2$Unif, Unif, closed under epimorphic images, or being algebraic

Abstract

The epireflective subcategories of ${\bold{Top}}$, that are closed under epimorphic (or bimorphic) images, are $\{X \mid |X| \le 1 \} $, $\{X \mid X$ is indiscrete$\} $ and ${\bold{Top}}$. The epireflective subcategories of ${\bold{T_2Unif}}$, 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 ${\bold{T_2Unif}}$. The epireflective subcategories of ${\bold{Unif}}$, 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 ${\bold{Unif}}$. The epireflective subcategories of ${\bold{Top}}$, that are algebraic categories, are $\{X \mid |X| \le 1 \} $, and $\{X \mid X$ is indiscrete$\} $. The subcategories of ${\bold{Unif}}$, 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 ${\bold{Unif}}$, 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.