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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.
MSC:
54B30 Categorical methods in general topology
54A05 Topological spaces and generalizations (closure spaces, etc.)
54D05 Connected and locally connected spaces (general aspects)
Citations:
Zbl 0676.54012
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