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On generalized topological spaces. I. (English) Zbl 1268.54002
In [H. Delfs and M. Knebusch, Locally semialgebraic spaces. Lecture Notes in Mathematics. 1173. Berlin etc.: Springer-Verlag. (1985; Zbl 0582.14006)], Delfs and Knebusch generalized classical topology with “a ‘topologie générale’ for semialgebraic geometry”. The corresponding generalized spaces originate in the categorical concept of a Grothendieck topology.
In the paper under review the author initiates a systematic study of the category GTS of generalized topological spaces and their strictly continuous mappings. The paper provides the original axiomatization and the basic theory of GTS. Connections with the concept of a bornological universe are found. The new concept of admissibility is explained and generalized topological concepts such as small sets, bases, connectedness and discreteness are treated. Further it is proved that both GTS and its full subategory of small spaces are topological constructs.

MSC:
54A05 Topological spaces and generalizations (closure spaces, etc.)
03C07 Basic properties of first-order languages and structures
46A17 Bornologies and related structures; Mackey convergence, etc.
06B99 Lattices
54B30 Categorical methods in general topology
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