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Quotients of countably based spaces are not closed under sobrification

Published online by Cambridge University Press:  17 May 2006

GARY GRUENHAGE  and
THOMAS STREICHER
GARY GRUENHAGE
Affiliation:
Department of Mathematics, Auburn University, Auburn, AL 36849-5310, U.S.A.
THOMAS STREICHER
Affiliation:
Fachbereich Mathematik, Technische Universität Darmstadt, Schloßgartenstr. 7, 64289 Darmstadt, Germany.
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Abstract

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In this note we show that quotients of countably based spaces (qcb spaces) and topological predomains, as introduced by M. Schröder and A. Simpson, are not closed under sobrification. As a consequence, replete topological predomains need not be sober, that is, in general, repletion is not given by sobrification. Our counterexample also shows that a certain tentative ‘equaliser construction’ of repletion fails for qcb spaces.

Our results also extend to the more general class of core compactly generated spaces.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 16 , Issue 2 , April 2006 , pp. 223 - 229
Copyright
2006 Cambridge University Press