Parovicenko space
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
A Parovicenko space is a topological space X satisfying the following conditions:
Properties
The space βN\N is a Parovicenko space, where βN is the Stone–Čech compactification of the natural numbers N. {{harvtxt|Parovicenko|1963}} proved that the continuum hypothesis implies that every Parovicenko space is isomorphic{{clarify|reason=what is meant here by "isomorphism"? homeomorphism?|date=January 2021}} to βN\N. {{harvtxt|van Douwen|van Mill|1978}} showed that if the continuum hypothesis is false then there are other examples of Parovicenko spaces.
References
- {{Cite journal|title=Parovicenko's Characterization of βω- ω Implies CH
|first1= Eric K. |last1=van Douwen|first2=Jan |last2=van Mill
|journal= Proceedings of the American Mathematical Society|volume=72|issue= 3|year=1978|pages= 539–541
|doi=10.2307/2042468|jstor=2042468}}
- {{cite journal|mr=0150732
|last=Parovicenko|first= I. I.
|title=[On a universal bicompactum of weight ℵ]
|journal=Doklady Akademii Nauk SSSR |volume=150 |year=1963 |pages=36–39}}