Epireflective subcategories of Top, $T_2$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 ${\mathbf{T}}_2\mathbf{Unif}$, 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 ${\mathbf{T}}_2\mathbf{Unif}$. 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.

20 pages