Parovicenko space

In mathematics, a Parovicenko space is a space similar to the space of non-isolated points of the Stone-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 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.

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 
• Parovicenko, I. I. (1963). "[On a universal bicompactum of weight ℵ]". Doklady Akademii Nauk SSSR 150: 36–39. MR0150732.