Parovicenko space
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In mathematics, a Parovicenko space is a space similar to the space of non-isolated points of the Stone-Cech compactification of the integers.
[edit] 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 Fσ subsets of X have disjoint closures
- Every nonempty Gδ of X has non-empty interior.
[edit] Properties
The space βN − N is a Parovicenko space, where βN is the Stone-Cech compactification of the natural numbers N. Parovicenko (1963) proved that the continuum hypothesis implies that every Parovicenko space is isomorphic to βN − N. van Douwen & van Mill (1978) showed that if the continuum hypothesis is false then there are other examples of Parovicenko spaces.
[edit] References
- van Douwen, Eric K. & van Mill, Jan (1978), “Parovicenko's Characterization of βω- ω Implies CH”, Proceedings of the American Mathematical Society 72 (3): 539-541, <http://links.jstor.org/sici?sici=0002-9939%28197812%2972%3A3%3C539%3APCOIC%3E2.0.CO%3B2-1>
- Parovicenko, I. I. (1963), “On a universal bicompactum of weight ℵ.”, Dokl. Akad. Nauk SSSR 150: 36-39, MR0150732
