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On hyperconnected topological spaces. (English) Zbl 1389.54041

Summary: It is proved that there is only one open ultrafilter iff the space is hyperconnected. Relations of functions demi-open, skeletal and o-ultra, obtained without and with regard to hyperconnectedness, are used to prove that the full subcategory HypTop of Top, consisting of all hyperconnected spaces, is a subcategory of the category skelTOP of all topological spaces with morphisms skeletal continuous functions. HypTop is proved to be mono-epireflective in skelTop by showing that there exists a functor from skelTOP to HypTop turning skelTop into the largest subcategory of Top where the functor is a reflection. Also, skelTop is found to be a maximal subcategory of TOP such that HypTop is reflective in the subcategory with reflection morphisms skeletal functions.

MSC:

54B30 Categorical methods in general topology
54C08 Weak and generalized continuity
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54G05 Extremally disconnected spaces, \(F\)-spaces, etc.
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