UDC 513.83

MATHEMATICS

A. I. RAÏKHBERG

EXTREMALLY DISCONNECTED SPACES WITH A UNIQUE HOMEOMORPHISM ONTO THEMSELVES

(Presented by Academician P. S. Aleksandrov on 23 IV 1970)

In 1948 G. Birkhoff \((^{1})\) posed the problem: does every Boolean algebra have a nonidentical isomorphism onto itself? In 1951 M. Katětov \((^{6})\) constructed a counterexample. The question remained unresolved for complete (m-complete) Boolean algebras, or in the equivalent formulation: does every extremally disconnected* (m-extremally disconnected) bicompactum have a nonidentical homeomorphism onto itself? If in this formulation bicompactness is abandoned, then the problem is solved negatively (Theorem 2).

All spaces in this note are completely regular. By \(pX\) we shall denote the absolute** of the topological space \(X\), by \(G(X)\) the group of homeomorphisms of \(X\) onto itself, and by \(G_x(X)\) the orbit of the point \(x \in X\) with respect to this group; by \(GF(X)\) and \(GFI(X)\) the groups of perfect multivalued and irreducible perfect multivalued mappings of \(X\) onto \(X\); by \(wX\), \(\pi X\), \(sX\), and \(cX\) the weight, \(\pi\)-weight, density, and Suslin number of the space \(X\), respectively.

We formulate a number of previously proved results that we shall need.

Proposition 1. Let \(X\) be an extremally disconnected bicompactum, and let \(D\) be strongly discrete in \(X\). Then \([D] = \beta D\) \((^{4,2})\).

Proposition 2. Let \(X\) be an extremally disconnected bicompactum, and let \(S\) be a dense subset of \(X\). Then \(S\) is extremally disconnected and \(\beta S = X\) \((^{2,10})\).

Proposition 3. Let \(f : X \to Y\) be a perfect irreducible (single-valued) mapping. Then \(\pi X = \pi Y\), \(cX = cY\) (moreover, the inverse image of a dense in \(Y\) system of open sets is a dense in \(X\) system of open sets) \((^{7,9})\).

Proposition 4. Let \(Y\) be an extremally disconnected bicompactum of \(\pi\)-weight \(\tau\). Then there exists a bicompactum \(X\) of weight \(\tau\), whose absolute is homeomorphic to \(Y\) \((^{7})\).

Proposition 5. \(GFI(X) \approx G(pX)\) \((^{5,8,10})\).

Proposition 6. In order that a multivalued mapping \(f : X \to Y\) be perfect, it is necessary and sufficient that there exist a space \(Z\) (the graph of the mapping \(f\)) and single-valued perfect mappings

\[ \varphi : Z \to X,\qquad \psi : Z \to Y,\quad \text{such that } f=\psi\varphi^{-1}. \]

Lemma. Let \(X\) be a bicompactum of weight \(\tau\). Then \(|GF(X)| \leq \exp \tau\).

Proof. Let \(f \in GF(X)\). It is determined (Proposition 6) by its graph \(Z \subseteq X \times X\) and the projections \(\varphi, \psi : Z \to X\), where \(Z\) is a bicompactum and \(wZ \leq \tau\), and there are \(\leq \exp \tau\) such bicompacta. Mappings

* The properties of extremally disconnected spaces are set out sufficiently fully in \((^{2})\).

** All notions connected with the theory of absolutes—irreducible perfect mappings, dense systems, \(\pi\)-weight—can be found in the papers \((^{3,5,7–10})\).

\(\varphi: Z \to X\) is also \(\leq \exp \tau\), since \(sZ \leq \tau\), \(|X| \leq \exp \tau\), and \((\exp \tau)^\tau = \exp(\tau^2)=\exp \tau\). Thus \(|GF(X)| \leq (\exp \tau)^3 = \exp \tau\).

Theorem 1. Let \(Y\) be an extremally disconnected bicompactum of \(\pi\)-weight \(\tau\). Then \(|G(Y)| \leq \exp \tau\); in particular, if \(|Y| > \exp \tau\), then \(Y\) is nonhomogeneous.

Proof. There exists a bicompactum \(X\) of weight \(\tau\), whose absolute is homeomorphic to \(Y\) (Proposition 4); then (Proposition 5 and the lemma) we obtain that
\[ |G(Y)|=|GF(X)|\leq GF(X)\leq \exp \tau . \]

Theorem 2. There exists an extremally disconnected space \(Y\) with a unique homeomorphism onto itself \(\operatorname{id}_Y\); moreover, it can be chosen so that
\[ |Y|=\pi Y=sY=cY=\tau, \]
where \(\tau\) is an arbitrary infinite cardinal.

Proof. Let \(D\) be a discrete space of cardinality \(\tau\), and let \(bD\) be its one-point bicompactification,
\[ X=(bD)^{\aleph_0}=\prod_{i=1}^{\infty}(bD)_i . \]

Observe that \(wX=\pi X=sX=cX=\tau\). We shall show that for every open set \(U\) in \(X\), \(cU=\tau\). Indeed, choose a number \(n\) such that \(\pi_n U=bD\) (\(\pi_n\) is the projection of \(X\) onto the \(n\)-th factor); then the disjoint system
\[ \left\{\,U\cap\left(\{x\}_n\times \prod_{i\ne n}(bD)_i\right):\ x\in D\,\right\} \]
of open nonempty sets will be the desired one. Consequently (Propositions 1 and 3), every open set in \(pX\) has cardinality \(\geq \exp\exp \tau\).

Observe that \(\pi pX=spX=cpX=\tau\) (Proposition 3); then \(|pX|=\exp\exp \tau\) and \(|U|=\exp\exp \tau\) for every open set \(U\) in \(pX\), and, consequently, \(wpX=\exp \tau\). Thus, there exist nonhomogeneous extremally disconnected bicompacta \(pX\) with \(\Delta pX=|pX|^*\), \(\pi pX=spX=cpX=\tau\), where \(\tau\) is an arbitrary infinite cardinal.

Let \(\{V_\xi:\xi<\omega,\ |\omega|=\tau\}\) be a dense system of open subsets of \(pX\), which contains a disjoint subsystem of cardinality \(\tau\). We successively choose points
\[ y_\xi\in V_\xi\setminus \bigcup\{G_{y_\eta}(pX):\eta<\xi\}, \]
which can be done, since
\[ |V_\xi|=\exp\exp \tau>\exp \tau\geq \left|\bigcup\{G_{y_\eta}(pX):\eta<\xi\}\right|. \]
Put
\[ Y=\{y_\xi:\xi<\omega\}; \]
observe that \([Y]=pX\), and therefore \(Y\) is extremally disconnected (Proposition 2). From the construction it follows that, first, \(G(Y)=\{\operatorname{id}_Y\}\) (Proposition 2), and, second,
\[ |Y|=\pi Y=sY=cY=\tau, \]
for
\[ \tau=|Y|\geq \pi Y\geq sY\geq cY\geq \tau . \]
The theorem is proved.

The work was written under the supervision of V. I. Ponomarev, to whom I express my deep gratitude.

Moscow State University
named after M. V. Lomonosov

Received
15 IV 1970

CITED LITERATURE

1. G. Birkhoff, Lattice Theory, N. Y., 1948, 1961; Теория структур, M., 1952.
2. L. Gillman, M. Jerison, Rings of Continuous Functions, Princeton, 1960.
3. A. M. Gleason, M. J. Illinois, 2, 4a, 482 (1958).
4. Б. Ефимов, ДАН, 183, No. 3, 511 (1968).
5. С. Илиадис, С. Фомин, УМН, 21, 4, 47 (1966).
6. M. Katětov, Coll. Math., 2, 229 (1951).
7. В. И. Пономарев, УМН, 21, 4, 91 (1966).
8. В. И. Пономарев, Матем. сборн., 60, 89 (1963).
9. В. И. Пономарев, Матем. сборн., 51, 515 (1960).
10. В. И. Пономарев, ДАН, 149, 26 (1963).

* The dispersion character \(\Delta X=\min\{|U|:\ U \text{ is open in } X\}\).