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Compact sets in non-metrizable product spaces

✍ Scribed by J. van der Slot

Publisher
Elsevier Science
Year
1972
Weight
680 KB
Volume
2
Category
Article
ISSN
0016-660X

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✦ Synopsis

Abstxact: Denote by s *thy countable infinite topological product of real B-W. Pi, result of Anderson and K.le@ states tit whenever C i9 a comp& subset of s, then s \ C is haraeomorphic with s. In this note we show that, for products of more than countabfiy many reaI Irines the &uation is completeIy different; in fact we have the fokMng P~~su~~: Let Cl and. 62 bz compact subsets of ati uncountable product P of real lines. Then P f Cl is Bomeom&pMc Wh pYW2 if %xd only if Cs is homeomorphic u&h C+J. Furthermore, 3ere exist two cloiped countable &crete &VP sets S1 and S, &f P such that P\ Sl is rtot homeomctrphic with P Wi.

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## Abstract We study the role that the axiom of choice plays in Tychonoff's product theorem restricted to countable families of compact, as well as, LindelΓΆf metric spaces, and in disjoint topological unions of countably many such spaces. (Β© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)