## Cell structures and topologically complete spaces.(English)Zbl 1390.54014

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 [W. Dębski and 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.
Reviewer: Ziga Virk (Litija)

### MSC:

 54C05 Continuous maps 54B35 Spectra in general topology 54D30 Compactness 54E15 Uniform structures and generalizations

Zbl 1382.54018
Full Text:

### References:

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