Cell structures and topologically complete spaces
The overarching theme encompassing this paper is that of approximations in topology: given a topological space, we would like to approximate it by simpler spaces. The idea goes back to Alexandroff who approximated compact metric spaces by polyhedra. Freudenthal proved that if \(X\) is a compact metric space then it is homeomorphic to the inverse limit of an inverse sequence of polyhedra whose dimension is bounded by the dimension of \(X\). These ideas developed into Shape Theory and later influenced similar approaches in the asymptotic setting (Anti-Čech approximations by Dranishnikov and the combinatorial approach by Dydak et al.). While approximations by complexes are in a way most natural, it is in principle possible to construct approximations by other spaces. For example, Kopperman et al. used finite \(T_0\) spaces in such approximations. In their paper [\textit{W. Dębski} and \textit{E. D. Tymchatyn}, Colloq. Math. 147, No. 2, 181--194 (2017; Zbl 1382.54018)], the authors introduced cell structures, which are approximations by combinatorial graphs, the latter being considered as discrete spaces. They showed that a complete metric space can be obtained from a cell structure by taking the appropriate quotient of the inverse limit. Furthermore, they showed that continuous maps between such spaces are determined by their approximations via a cell structure. In this paper the authors generalize these results to topologically complete spaces (and continuous maps between them). In contrary to approximations by complexes, a topologically complete space is only a quotient of the obtained inverse limit of a cell structure. On the other hand, cell structures turn out to be technically much more manageable than approximations by complexes.
scientific article; zbMATH DE number 6853367