Almost reflective subcategories of \(\mathbf T \mathbf o \mathbf p\)
The paper deals with the relationship between almost reflective subcategories (= weakly reflective plus closed under the formation of retracts) and implicational subcategories (= injectivity classes in the sense of [\textit{J. Adámek} and \textit{J. Rosický}, Trans. Am. Math. Soc. 336, 785-804 (1993; Zbl 0789.18003)]), predominantly in the catgory \(\underline {\text{Top}}\) of topological spaces. Implicational subcategories of \(\underline {\text{Top}}\) are exactly the intersections of almost reflective subcategories, just as orthogonal subcategories are the intersections of reflective subcategories [cf. \textit{J. Rosický} and the reviewer, Cah. Topologie Géom. Différ. Catégoriques 29, No. 3, 203-215 (1988; Zbl 0669.18001)]. Typical results are: all (Hausdorff, Tikhonov, resp.) spaces form the implicational hull of connected (\(H\)- closed, pseudocompact, resp.) spaces; in the ordered conglomerate of all almost reflective subcategories of \(\underline {\text{Top}}\), binary infima may fail to exist; \(\underline {\text{Top}}\) has no productive almost coreflective subcategory consisting of \(T_ 0\)-spaces only.
scientific article; zbMATH DE number 221868
scientific article