Parovicenko space
(Redirected from Parovicenko's theorem)
In mathematics, a Parovicenko space is a topological space similar to the space of non-isolated points of the Stone–Čech compactification of the integers.
Definition
[edit]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 non-empty Gδ of X has non-empty interior.
Properties
[edit]The space βN\N is a Parovicenko space, where βN is the Stone–Čech compactification of the natural numbers N. Parovicenko (1963) proved that the continuum hypothesis implies that every Parovicenko space is isomorphic[clarification needed] to βN\N. van Douwen & van Mill (1978) showed that if the continuum hypothesis is false then there are other examples of Parovicenko spaces.
References
[edit]- 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. doi:10.2307/2042468. JSTOR 2042468.
- Parovicenko, I. I. (1963). "[On a universal bicompactum of weight ℵ]". Doklady Akademii Nauk SSSR. 150: 36–39. MR 0150732.