On a problem of Herrlich concerning topological axioms (Q1071105)

Each topological category is concretely isomorphic to a reflective modification of some functor-structured category \(S(F)\). \textit{H. Herrlich} introduced a notion of topological axioms and proved from the above fact that each topological category is definable by a class \(\Gamma\) of topological axioms in \(S(F)\) [Categorical topology, Proc. int. Conf., Toledo/Ohio 1983, Sigma Ser. Pure Math. 5, 223--281 (1984; Zbl 0548.18004)]. In this paper the author gives a necessary condition in order that a topological category \(\mathcal K\) has a small presentation \(\Gamma\), i.e. \(\mathcal K\) is definable by a set \(\Gamma\) of topological axioms and he answers the question put by Herrlich as follows. The category \(\mathbf{Top}\) of topological spaces has no small presentation in any functor-structured category. He also shows that the above condition is insufficient and hence the problem of a characterization of topological categories having a small presentation is left still open.