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On supercomplete uniform spaces. IV: Countable products. (English) Zbl 0713.54027
[For part I-II see Zbl 0538.54015, Zbl 0654.54020.]
Generally the product of paracompact spaces is not paracompact, but it is known that it is true for the countable product of locally compact paracompact spaces and for paracompact p-spaces. The author proves that the countable product of supercomplete C-scattered spaces is supercomplete. Supercompleteness was introduced by J. R. Isbell in Pac. J. Math. 12, 287-290 (1962; Zbl 0104.395) and can be considered as a uniform version of paracompactness. A uniform space $$\mu$$ X is called supercomplete if the hyperspace of closed subsets H($$\mu$$ X) equipped with the Hausdorff uniformity is a complete uniform space.
Reviewer: R.Welk

##### MSC:
 54E15 Uniform structures and generalizations 54B10 Product spaces in general topology 54B20 Hyperspaces in general topology 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54G12 Scattered spaces
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