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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)
Zbl 0676.54012
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