- Parovicenko space
In
mathematics , a Parovicenko space is a space similar to the space of non-isolated points of theStone-Cech compactification of the integers.Definition
A Parovicenko space is a topological space "X" satisfying the following conditions:
*"X" is compact Hausdorff
*"X" has no isolated points
*"X" has weight "c", the cardinality of the continuum (this is the smallest cardinality of a base for the topology).
*Every two disjoint open "F"σ subsets of "X" have disjoint closures
*Every nonempty "G"δ of "X" has non-empty interior.Properties
The space β"N" − "N" is a Parovicenko space, where β"N" is the
Stone-Cech compactification of the natural numbers "N". harvtxt|Parovicenko|1963 proved that thecontinuum hypothesis implies that every Parovicenko space is isomorphic to β"N" − "N". harvtxt|van Douwen|van Mill|1978 showed that if the continuum hypothesis is false then there are other examples of Parovicenko spaces.References
*citation|title=Parovicenko's Characterization of βω- ω Implies CH
first= Eric K. |last=van Douwen|first2=Jan |last2=van Mill
journal= Proceedings of the American Mathematical Society|volume=72|issue= 3|year=1978|pages= 539–541
url= http://links.jstor.org/sici?sici=0002-9939%28197812%2972%3A3%3C539%3APCOIC%3E2.0.CO%3B2-1|doi=10.2307/2042468
*citation|id=MR|0150732
last=Parovicenko|first= I. I.
title=On a universal bicompactum of weight ℵ.
journal=Dokl. Akad. Nauk SSSR |volume=150 |year=1963 |pages=36–39
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