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

The epireflective subcategories of Top, that are closed under epimorphic (or bimorphic) images, are { X∣ | X| ≤ 1 } , { X∣ X is indiscrete} and Top. The epireflective subcategories of T2Unif, closed under epimorphic images, are: { X∣ | X| ≤ 1 } , { X∣ X is compact T2} , { X∣ covering character of X is ≤ λ0} (where λ0 is an infinite cardinal), and T2Unif. The epireflective subcategories of Unif, closed under epimorphic (or bimorphic) images, are: { X∣ | X| ≤ 1 } , { X∣ X is indiscrete} , { X∣ covering character of X is ≤ λ0} (where λ0 is an infinite cardinal), and Unif. The epireflective subcategories of Top, that are algebraic categories, are { X∣ | X| ≤ 1 } , and { X∣ X is indiscrete}. The subcategories of Unif, closed under products and closed subspaces and being varietal, are { X∣ | X| ≤ 1 } , { X∣ X is indiscrete} , { X∣ X is compact T2}. The subcategories of Unif, closed under products and closed subspaces and being algebraic, are { X∣ X is indiscrete} , and all epireflective subcategories of { X∣ X is compact T2}. Also we give a sharpened form of a theorem of Kannan-Soundararajan about classes of T3 spaces, closed for products, closed subspaces and surjective images. © 2016, Akadémiai Kiadó, Budapest, Hungary