Parovicenko space

In mathematics, a Parovicenko space is a space similar to the space of non-isolated points of the Stone–Čech 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 Fσ subsets of X have disjoint closures
  • Every nonempty Gδ of X has non-empty interior.

Properties

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 to βN\N. van Douwen & van Mill (1978) showed that if the continuum hypothesis is false then there are other examples of Parovicenko spaces.

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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. doi:10.2307/2042468. JSTOR 2042468.CS1 maint: ref=harv (link)
  • Parovicenko, I. I. (1963). "[On a universal bicompactum of weight ℵ]". Doklady Akademii Nauk SSSR. 150: 36–39. MR 0150732.CS1 maint: ref=harv (link)
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