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\(\beta\)-connectedness and \(\mathcal S\)-connectedness of topological spaces. (English) Zbl 1289.54004

The author considers the following open-like sets in a topological space \(X\): \(\alpha\)-open, semi-open, preopen, \(b\)-open and \(\beta\)-open = semi-preopen, and the corresponding closure operators. Different notions of connectedness are defined when in the usual definition of connectedness, open sets are replaced with two members from one or two of the above collections of open-like sets. Characterizations of these notions of connectedness are given by means of closure operators and it is proved that some types coincide. Preservation under surjections of these connectedness-like properties is studied as well.

MSC:

54A05 Topological spaces and generalizations (closure spaces, etc.)
54D05 Connected and locally connected spaces (general aspects)
54C08 Weak and generalized continuity
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