View source for Cech-Stone compactification of omega

The [[Stone space|Stone space]] of the [[Boolean algebra|Boolean algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c1201002.png" />. It is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c1201003.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c1201004.png" /> is the power set algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c1201005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c1201006.png" /> denotes its ideal of finite sets. The points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c1201007.png" /> can be identified with ultrafilters (cf. [[Ultrafilter|Ultrafilter]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c1201008.png" />. There are two types of ultrafilters: fixed ultrafilters, which correspond to the points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c1201009.png" />, and free ultrafilters, which correspond to the points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010010.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010011.png" /> has attracted quite a lot of attention the last decades (1998). See e.g. [[#References|[a10]]], [[#References|[a4]]], and the survey paper [[#References|[a15]]].

The Parovichenko theorem from 1963 (see [[#References|[a16]]] and [[Parovichenko algebra|Parovichenko algebra]]) asserts that under the [[Continuum hypothesis|continuum hypothesis]] (abbreviated CH), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010012.png" /> is topologically the unique zero-dimensional compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010014.png" />-space of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010015.png" /> (the cardinality of the continuum, cf. also [[Continuum, cardinality of the|Continuum, cardinality of the]]) in which non-empty <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010016.png" /> sets have infinite interior (for short, a Parovichenko space). This theorem had wide applications both in topology as well as in the theory of Boolean algebras. The method of proof of this result goes back to W. Rudin [[#References|[a18]]]. In 1978, E.K. van Douwen and J. van Mill proved the converse to the Parovichenko theorem [[#References|[a6]]]: If all Parovichenko spaces are homeomorphic, then the continuum hypothesis holds.

The Parovichenko theorem implies that every compact space of weight at most the continuum is a continuous image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010017.png" /> under CH. This result cannot be proved in ZFC alone (cf. also [[Set theory|Set theory]]). In 1968, K. Kunen [[#References|[a12]]] proved that in a model formed by adding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010018.png" /> Cohen reals to a model of the continuum hypothesis, there is no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010019.png" /> sequence of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010020.png" /> which is strictly decreasing (modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010021.png" />). This implies that the ordinal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010022.png" /> is not a continuous image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010023.png" />. In 1978, E.K. van Douwen and T.C. Przymusiński [[#References|[a7]]] used results of F. Rothberger [[#References|[a17]]] to prove that there is a counterexample under the hypothesis

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010024.png" /></td> </tr></table>

This is interesting since it only involves a hypothesis on cardinal numbers.

W. Rudin [[#References|[a18]]] proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010025.png" /> is not topologically homogeneous under CH, by establishing two different types of points with obvious distinct topological behaviour: the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010026.png" />-points and the non-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010027.png" />-points. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010029.png" />-point in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010030.png" /> is a point having the property that the intersection of any countable family of its neighbourhoods is again a neighbourhood. It was shown later by S. Shelah in [[#References|[a22]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010031.png" />-points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010032.png" /> need not exist in ZFC alone. That <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010033.png" /> is not homogeneous is a theorem of ZFC, however; it was shown by Z. Frolik [[#References|[a9]]]. Rudin [[#References|[a18]]] also proved that under CH there is only one type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010034.png" />-point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010035.png" />, in the sense that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010036.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010037.png" />-points, then there is an auto-homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010038.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010039.png" />. The method of this proof was used later by I.I. Parovichenko (see above).

Frolik's proof of the inhomogeneity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010040.png" /> does not produce two points with obvious distinct topological behaviour. Call a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010041.png" /> in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010042.png" /> a weak <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010044.png" />-point if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010045.png" /> is not in the closure of any countable subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010046.png" />. The last and final word on the inhomogeneity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010047.png" /> came from K. Kunen [[#References|[a14]]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010048.png" /> contains weak <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010049.png" />-points.

B. Balcar and P. Vojtáš [[#References|[a1]]] proved that every point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010050.png" /> is simultaneously in the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010051.png" />-many pairwise disjoint non-empty open subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010052.png" />.

S. Shelah [[#References|[a19]]] proved it to be consistent that all auto-homeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010053.png" /> are induced by a partial permutation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010054.png" />. By a partial permutation one understands a bijective function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010055.png" />, where both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010057.png" /> are [[cofinite subset]]s of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010058.png" />. So, consequently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010059.png" /> has only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010060.png" />-many auto-homeomorphisms (it is easy to see that under CH, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010061.png" /> has precisely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010062.png" />-many auto-homeomorphisms). Shelah's result was used later by E.K. van Douwen [[#References|[a5]]] to show that the auto-homeomorphism group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010063.png" /> need not be algebraically simple. So, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010064.png" /> is a homogeneous Boolean algebra whose automorphism group need not be algebraically simple.

There are various interesting partial orders on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010065.png" />. The most well-known and the most useful is the so-called Rudin–Keisler order, which is defined as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010066.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010067.png" /> if there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010068.png" /> which sends <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010069.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010070.png" />. Kunen [[#References|[a13]]] proved that there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010071.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010073.png" />. This result implies that no infinite compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010074.png" />-space is homogeneous, as well as various other interesting and non-trivial facts. See [[#References|[a3]]] for more details. In fact, Kunen [[#References|[a13]]] proved that there is a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010075.png" /> of size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010076.png" /> which consists of pairwise Rudin–Keisler-incomparable points. This result was subsequently strengthened by Shelah [[#References|[a20]]]: there even exists such a set of size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010077.png" />. A more general theorem, with a simpler proof, was later proved by A. Dow [[#References|[a8]]].

There is also an interesting [[Semi-group|semi-group]] structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010078.png" />. S. Glazer (see [[#References|[a3]]]) used the existence of idempotents in the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010079.png" /> to give a particularly simple topological proof of Hindman's theorem from [[#References|[a11]]]: If the set of natural numbers is divided into two sets, then there is a sequence drawn from one of these sets such that all finite sums of distinct numbers of this sequence remain in the same set.

Several other results from classical number theory can be proved as well by similar methods. In [[#References|[a2]]], V. Bergelson, H. Furstenberg, N. Hindman, and Y. Katznelson again used the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010080.png" /> to present an elementary proof of van der Waerden's theorem from [[#References|[a21]]]: If the natural numbers are partitioned into finitely many classes in any way whatever, one of these classes contains arbitrarily long arithmetic progressions.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Balcar,   P. Vojtáš,   "Almost disjoint refinement of families of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010081.png" />"  ''Proc. Amer. Math. Soc.'' , '''79'''  (1980)  pp. 465–470</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V. Bergelson,   H. Furstenberg,   N. Hindman,   Y. Katznelson,   "An algebraic proof of van der Waerden's theorem"  ''Enseign. Math.'' , '''35'''  (1989)  pp. 209–215</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W.W. Comfort,   "Ultrafilters: some old and some new results"  ''Bull. Amer. Math. Soc.'' , '''83'''  (1977)  pp. 417–455</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W.W. Comfort,   S. Negrepontis,   "The theory of ultrafilters" , ''Grundl. Math. Wissenschaft.'' , '''211''' , Springer  (1974)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E.K. van Douwen,   "The automorphism group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010082.png" /> need not be simple"  ''Topol. Appl.'' , '''34'''  (1990)  pp. 97–103</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  E.K. van Douwen,   J. van Mill,   "Parovičenko's characterization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010083.png" /> implies CH"  ''Proc. Amer. Math. Soc.'' , '''72'''  (1978)  pp. 539–541</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  E.K. van Douwen,   T.C. Przymusiński,   "Separable extensions of first countable spaces"  ''Fund. Math.'' , '''105'''  (1980)  pp. 147–158</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  Alan Dow,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010084.png" />: The work of Mary Ellen Rudin"  ''Ann. New York Acad. Sci.'' , '''705'''  (1993)  pp. 47–66</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  Z. Frolik,   "Sums of ultrafilters"  ''Bull. Amer. Math. Soc.'' , '''73'''  (1967)  pp. 87–91</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  L. Gillman,   M. Jerison,   "Rings of continuous functions" , v. Nostrand  (1960)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  N. Hindman,   "Finite sums from sequences within cells of a partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010085.png" />"  ''J. Combin. Th. A'' , '''17'''  (1974)  pp. 1–11</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  K. Kunen,   "Inaccessibility properties of cardinals"  ''PhD Thesis Stanford Univ.''  (1968)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  K. Kunen,   "Ultrafilters and independent sets"  ''Trans. Amer. Math. Soc.'' , '''172'''  (1972)  pp. 299–306</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  K. Kunen,   "Weak P-points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010086.png" />" , ''Topology, Colloq. Math. Soc. János Bolyai'' , '''23'''  (1980)  pp. 741–749</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  J. van Mill,   "An introduction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010087.png" />"  K. Kunen (ed.)  J.E. Vaughan (ed.) , ''Handbook of Set–Theoretic Topology'' , North-Holland  (1984)  pp. 503–567</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  I.I. Parovičenko,   "A universal bicompact of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010088.png" />"  ''Soviet Math. Dokl.'' , '''4'''  (1963)  pp. 592–592</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  F. Rothberger,   "A remark on the existence of a denumberable base for a family of functions"  ''Canad. J. Math.'' , '''4'''  (1952)  pp. 117–119</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  W. Rudin,   "Homogeneity problems in the theory of Čech compactifications"  ''Duke Math. J.'' , '''23'''  (1956)  pp. 409–419</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  S. Shelah,   "Proper forcing" , ''Lecture Notes Math.'' , '''940''' , Springer  (1982)</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  S. Shelah,   M.E. Rudin,   "Unordered types of ultrafilters"  ''Topol. Proc.'' , '''3'''  (1978)  pp. 199–204</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top">  B. van der Waerden,   "Beweis einer Baudetschen Vermutung"  ''Nieuw Arch. Wiskunde'' , '''19'''  (1927)  pp. 212–216</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top">  E. Wimmers,   "The Shelah <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120100/c12010089.png" />-point independence theorem"  ''Israel J. Math.'' , '''43'''  (1982)  pp. 28–48</TD></TR></table>