Cech-Stone compactification of omega

The Stone space of the Boolean algebra $\mathcal{P}(\omega)/\text{fin}$. It is denoted by $\beta\omega$. Here, $\mathcal{P}(\omega)$ is the power set algebra of $\omega$ and $\text{fin}$ denotes its ideal of finite sets. The points in $\beta\omega$ can be identified with ultrafilters on $\omega$. There are two types of ultrafilters: fixed ultrafilters, which correspond to the points in $\omega$, and free ultrafilters, which correspond to the points in $\omega^* = \beta\omega \setminus \omega$. The space $\omega^*$ has attracted quite a lot of attention the last decades (1998). See e.g. [a10], [a4], and the survey paper [a15].

The Parovichenko theorem from 1963 (see [a16] and Parovichenko algebra) asserts that under the continuum hypothesis (abbreviated CH), $\omega^*$ is topologically the unique zero-dimensional compact $F$-space of weight $\mathfrak{c}$ (cardinality of the continuum) in which non-empty $G_\delta$ 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 [a18]. In 1978, E.K. van Douwen and J. van Mill proved the converse to the Parovichenko theorem [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 $\omega^*$ under CH. This result cannot be proved in ZFC alone (cf. also Set theory). In 1968, K. Kunen [a12] proved that in a model formed by adding $\omega_2$ Cohen reals to a model of the continuum hypothesis, there is no $\omega_2$ sequence of subsets of $\omega$ which is strictly decreasing (modulo $\text{fin}$). This implies that the ordinal space $W(\mathfrak{c}+1)$ is not a continuous image of $\omega^*$. In 1978, E.K. van Douwen and T.C. Przymusiński [a7] used results of F. Rothberger [a17] to prove that there is a counterexample under the hypothesis $$ \omega_2 \le \mathfrak{c} \le 2^{\omega_1} = \omega_{\omega_2} \ . $$

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

W. Rudin [a18] proved that $\omega^*$ is not topologically homogeneous under CH, by establishing two different types of points with obvious distinct topological behaviour: the so-called $P$-points and the non-$P$-points. A $P$-point in a topological space $X$ 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 [a22] that $P$-points in $\omega^*$ need not exist in ZFC alone. That $\omega^*$ is not homogeneous is a theorem of ZFC, however; it was shown by Z. Frolik [a9]. Rudin [a18] also proved that under CH there is only one type of $P$-point in $\omega^*$, in the sense that if $x,y \in \omega^*$ are $P$-points, then there is an auto-homeomorphism $f : \omega^* \rightarrow \omega^*$ such that $f(x) = y$. The method of this proof was used later by I.I. Parovichenko (see above).

Frolik's proof of the inhomogeneity of $\omega^*$ does not produce two points with obvious distinct topological behaviour. Call a point $x$ in a topological space $X$ a weak $P$-point if $x$ is not in the closure of any countable subset of $X \setminus \{x\}$. The last and final word on the inhomogeneity of $\omega^*$ came from K. Kunen [a14]: $\omega^*$ contains weak $P$-points.

B. Balcar and P. Vojtáš [a1] proved that every point in $\omega^*$ is simultaneously in the closure of $\mathfrak{c}$-many pairwise disjoint non-empty open subsets of $\omega^*$.

S. Shelah [a19] proved it to be consistent that all auto-homeomorphism of $\omega^*$ are induced by a partial permutation of $\omega$. By a partial permutation one understands a bijective function $f : E \rightarrow F$, where both $E$ and $F$ are cofinite subsets of $\omega$. So, consequently, $\omega^*$ has only $\mathfrak{c}$-many auto-homeomorphisms (it is easy to see that under CH, $\omega^*$ has precisely $2^{\mathfrak{c}}$-many auto-homeomorphisms). Shelah's result was used later by E.K. van Douwen [a5] to show that the auto-homeomorphism group of $\omega^*$ need not be algebraically simple. So, $\mathcal{}P\omega/\text{fin}$ is a homogeneous Boolean algebra whose automorphism group need not be algebraically simple.

There are various interesting partial orders on $\beta\omega$. The most well-known and the most useful is the so-called Rudin–Keisler order, which is defined as follows: If $p,q \in \beta\omega$, then $p \le q$ if there exists a function $f:\omega\rightarrow\omega$ which sends $p$ to $q$. Kunen [a13] proved that there exist $p,q \in \beta\omega$ such that $p \not\le q$ and $q \not\le p$. This result implies that no infinite compact $F$-space is homogeneous, as well as various other interesting and non-trivial facts. See [a3] for more details. In fact, Kunen [a13] proved that there is a subset $A \subseteq\omega^*$ of size $\mathfrak{c}$ which consists of pairwise Rudin–Keisler-incomparable points. This result was subsequently strengthened by Shelah [a20]: there even exists such a set of size $2^{\mathfrak{c}}$. A more general theorem, with a simpler proof, was later proved by A. Dow [a8].

There is also an interesting semi-group structure on $\beta\omega$. S. Glazer (see [a3]) used the existence of idempotents in the semi-group $(\beta\omega,{+})$ to give a particularly simple topological proof of Hindman's theorem from [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 [a2], V. Bergelson, H. Furstenberg, N. Hindman, and Y. Katznelson again used the semi-group $(\beta\omega,{+})$ to present an elementary proof of van der Waerden's theorem from [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

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