In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, 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
  F-sigma set
  In mathematics, an Fσ set is a countable union of closed sets. The notation originated in France with F for fermé and σ for somme ....
  
   subsets of X have disjoint closures
• Every nonempty G_δ
  G-delta set
  In the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets. The notation originated in Germany with G for Gebiet meaning open set in this case and δ for Durchschnitt .The term inner limiting set is also used...
  
   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. proved that the continuum hypothesis

Continuum hypothesis

In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...

implies that every Parovicenko space is isomorphic to βN − N. showed that if the continuum hypothesis is false then there are other examples of Parovicenko spaces.