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Each closed subset of metric space \(X\) with \(\text{Ind} X=0\) is a retract. (English) Zbl 0827.54011

Summary: The main result is an extension theorem which says that every continuous map, from a closed \(G_\delta\) subset of a paracompact space \(X\) with \(0 = \text{Ind }X\) (“Ind” is the large inductive dimension) into a developable space, has a continuous extension to all of \(X\). A corollary is that each closed subset of metric space \(X\) with \(\text{Ind }X = 0\) is a retract. These results extend work of Robert L. Ellis [Proc. Am. Math. Soc. 30, 599-602 (1971; Zbl 0203.557)] when the range space is a complete metric space. We also present a metrization theorem: every countable compact space with a continuous ultrametric is metrizable.

MSC:

54C20 Extension of maps
54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
54F45 Dimension theory in general topology