A generalization of Herrlich's question on almost reflective and coreflective subclasses of Top and Unif (Q1840743)

H. Herrlich posed the question whether there are nontrivial classes of topological spaces that are almost reflective and almost coreflective at the same time. This paper investigates a modified question: when a nontrivial generalized reflective class of topological or uniform spaces is equivalent to a generalized coreflective class of spaces. The main results of the paper are: Theorem 1. Let \(A\) and \(B\) be equivalent nontrivial subclasses of TOP. Then \(A\) coincides with \(B\) in each of the following cases: (1) The Sierpinski space \(\{0,1\}\) with \(\{0\}\) open is in \(A\cup B\); (2) \(A\) is orthogonal containing the discrete spaces: (3) \(A\) is orthogonal and \(B\) is projective; (4) \(A\) is orthogonal and \(B\) is almost multicoreflective; (5) \(A\) is injective and \(B\) is coreflective. Theorem 2. If a nontrivial orthogonal subcategory of UNIF is equivalent to a multicoreflective subcategory of UNIF then these subcategories coincide.