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Cantor-connectedness revisited. (English) Zbl 0782.54010
In [Math. Nachr. 141, 183-226 (1989; Zbl 0676.54012)] the author has introduced approach spaces as a common generalization of topological spaces and of metric spaces. In particular the category \({\mathbf AP}\) of approach spaces is a topological construct that contains (a) the category \(\mathbf{Top}\) of topological spaces and continuous maps as a simultaneously bireflective and bicoreflective full (!) subcategory and (b) the category \(\mathbf{PMet}\) of pseudometric spaces and nonexpansive maps as a bicoreflective full (!) subcategory. In the present paper the author singles out a class \({\mathcal E}\) of approach spaces and defines an approach space to be connected provided it is \({\mathcal E}\)-connected in the sense of Preuss. Main results: (1) Products of connected spaces are connected. (2) A topological space is connected if and only if, considered as approach space, it is connected. (3) A pseudometric space is uniformly connected (= Cantor-connected) if and only if, considered as approach space, it is connected. The author then associates with each approach space \(X\) a number \(\text{con}(X)\) in \([0,\infty]\) as a measure of connectedness of \(X\). Results: (4) \(X\) is connected if and only if \(\text{con}(X)=0\). (5) \(\text{con}\bigl( \prod X_ i \bigr)=\text{Sup con}(X_ i)\) for families of nonempty approach spaces. (6) \(\text{con}(f[X])\leq\text{con}(X)\) for morphisms \(f:X\to Y\) in \(\mathbf{AP}\). (7) \(\text{con} \bigl( \bigcup X_ i\bigr) \leq \sup \text{con}(X_ i)\) for families of subspaces of \(X\) with nonempty intersection.
54B30 Categorical methods in general topology
54A05 Topological spaces and generalizations (closure spaces, etc.)
54D05 Connected and locally connected spaces (general aspects)
Zbl 0676.54012
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