UDC 513.83

MATHEMATICS

A. V. ARKHANGELSKII

AN EXTREMALLY DISCONNECTED BICOMPACTUM OF WEIGHT \(c\) IS NONHOMOGENEOUS*

(Presented by Academician P. S. Aleksandrov on 14 X 1966)

Recall that a bicompactum is called extremally disconnected if the closure of each of its open subsets is open (see \((^1)\)). Below further information is given on the structure of extremally disconnected bicompacta. The main results are Theorems 1 and 2.

Proposition 1. Every infinite bicompact subspace of an extremally disconnected space contains a subspace homeomorphic to the Čech extension of the natural numbers (denoted below by \(\beta N\)).

Proof. Every infinite bicompactum contains a countable discrete subspace. Therefore it suffices to prove Lemma 1.

Lemma 1. The closure \([A]\) of an infinite countable discrete subspace \(A\), contained in a bicompact subspace of an extremally disconnected space \(X\), is homeomorphic to \(\beta N\).

Obviously, \([A]\) is a bicompact extension of the space \(A\). Let \(A_1 \subset A\), \(A_2 \subset A\), and \(A_1 \cap A_2 = \Lambda\). In view of the countability of the sets \(A_1\) and \(A_2\) and the discreteness of \(A\), the sets \(A_1\) and \(A_2\) have in the space \(X\) disjoint open neighborhoods \(\mathcal U_1\) and \(\mathcal U_2\). We have \([\mathcal U_2] \cap \mathcal U_1 = \Lambda\), moreover \([\mathcal U_2] \supseteq A_2\) and \(\mathcal U_1 \supseteq A_2\). By virtue of the extremal disconnectedness of the space \(X\), the set \([\mathcal U_2]\) is open. Hence \([A_1] \cap [A_2] = \Lambda\), whence it follows that \([A]\) is homeomorphic to \(\beta N\).

Corollary 1. There does not exist an extremally disconnected bicompactum of cardinality \(c\).*

Corollary 2. The cardinality of an arbitrary infinite closed subspace of an extremally disconnected bicompactum is not less than \(2^c\).

Problem. Characterize those bicompacta which are homeomorphic to subspaces of extremally disconnected bicompacta.

Let \(X\) be a fixed topological space. By a pair we shall mean an expression of the form \((x,D)\), where \(D\) is some countable discrete subspace of the space \(X\), and \(x \in X\) is a point from the closure of the set \(D\) (not belonging to \(D\)). The pairs \((x,D)\), \((x',D')\) are called equivalent, \((x,D) \approx (x',D')\), if the spaces \(\{x\} \cup D\) and \(\{x'\} \cup D'\) are homeomorphic (the sets \(\{x\} \cup D\) and \(\{x'\} \cup D'\) are supposed to be endowed with the topology induced from \(X\)). A pair \((x^*,D^*)\) will be called a subpair of the pair \((x,D)\) (notation \((x^*,D^*) < (x,D)\)), if \(x=x^*\) and \(D^* \subseteq D\). The pairs \((x,D)\) and \((x',D')\) are called weakly equivalent (written as \((x,D) \sim (x',D')\)) if they have equivalent subpairs.

Proposition 2. Let \(X\) be a separable extremally disconnected space and let \(x_0 \in X\) be any point. Denote by \(\mathfrak F\) the family—*

* All spaces considered below are assumed in advance to be regular \(T_1\)-spaces.
* By \(c\) throughout this paper is denoted the cardinality of the continuum.
** Proposition 2 generalizes the corresponding assertion of Z. Frolík on \(\beta N\), whose proof was kindly communicated by him to the author and formed the basis of the argument given below.

of all pairs of the form \((x_0,D)\), and let \(\mathscr E\) be any subfamily of the family \(\mathscr F\), consisting of pairwise not weakly equivalent pairs. Then the cardinality of the family \(\mathscr E\) does not exceed \(c\).

Proof. To each pair \((x_0,D)\in\mathscr E\) assign, in some way, a countable disjoint cover \(\gamma(x_0,D)\) of the set \(D\) by open-and-closed subsets of \(X\). Denote by \(S\) some countable everywhere dense subset of \(X\). It is clear that the system \(\gamma(x_0,D)\) is uniquely recovered from the system \(\gamma'(x_0,D)\), formed by the intersections of the elements of the system \(\gamma(x_0,D)\) with the set \(S\). The cardinality of all distinct systems \(\gamma'(x_0,D)\) does not exceed \(c\), since the cardinality of the set of all possible partitions of a countable set into a countable collection of pairwise disjoint sets is equal to \(c\).

Suppose that the cardinality of the family \(\mathscr E\) is greater than \(c\). Then there are two pairs \((x_0,D_1)\), \((x_0,D_2)\) in \(\mathscr E\) for which \(\gamma'(x_0,D_1)=\gamma'(x_0,D_2)\). In that case also \(\gamma(x_0,D_1)=\gamma(x_0,D_2)\). Since \((x_0,D_1)\nsim (x_0,D_2)\), \([D_1\cap D_2]\ne x_0\). Each element of the system \(\gamma(x_0,D_1)=\gamma(x_0,D_2)\) meets the set \(D_1\) in exactly one point and the set \(D_2\) in exactly one point. Those elements of the system \(\gamma\) for which these points do not coincide we enumerate, in some way, in a sequence \(\mathcal U_1,\mathcal U_2,\ldots,\mathcal U_n,\ldots\), and put \(x_n=\mathcal U_n\cap D_1,\ y_n=\mathcal U_n\cap D_2\).

By what has been said, \(x_n\ne y_n,\ n=1,2,\ldots,\ldots\infty\). From \([D_1\cap D_2]\ne x_0\) it follows that
\[ x_0\in\left[\bigcup_{n=1}^{\infty}\{x_n\}\right] \quad\text{and}\quad x_0\in\left[\bigcup_{n=1}^{\infty}\{y_n\}\right]. \]
For each \(n\) choose, around the points \(x_n,y_n\), disjoint open neighborhoods \(Ox_n, Oy_n\), contained in \(\mathcal U_n\).

Then: 1)
\[ \left[\bigcup_{n=1}^{\infty} Ox_n\right]\ni x_0; \]
2)
\[ \left[\bigcup_{n=1}^{\infty} Ox_n\right]\cap\left\{\bigcup_{n=1}^{\infty} Oy_n\right\}; \]
3)
\[ \left[\bigcup_{n=1}^{\infty} Oy_n\right]\ni x_0. \]
These three relations are in contradiction with the extremal disconnectedness of the space \(X\). Proposition 2 is proved.

Let now \(P\) be an arbitrary bicompact infinite subspace of the extremally disconnected space \(X\), \(A\) some countable discrete subspace of the space \(P\) (fixed in what follows), and \(x_0\) a point of the closure of the set \(A\), \(x_0\notin A\). Consider the family \(\mathscr F\) of all \((x_0,D)\), where \(D\subseteq P\), and let \(\mathscr E\) be such a subfamily of the family \(\mathscr F\) that: 1) every pair in \(\mathscr F\) is weakly equivalent to some pair in \(\mathscr E\); 2) any two distinct pairs in \(\mathscr E\) are not weakly equivalent. A family \(\mathscr E\) with the indicated properties exists by virtue of the axiom of choice. From Proposition 2 it follows that the cardinality of the family \(\mathscr E\) does not exceed \(c\). Consider all subpairs of pairs belonging to \(\mathscr E\); their totality, whose cardinality is obviously equal to \(c\), we denote by \(\mathscr B\). Finally, denote by \(Q\) the family of those pairs \((y,D)\) such that: 1) \(y\in[A]\setminus A\); 2) \(D\subseteq A\); 3) the pair \((y,D)\) is equivalent to some pair belonging to the family \(\mathscr B\).

Proposition 3. The cardinality of the family \(Q\) is not greater than \(c\).

Proof. For each pair \((y,D)\in Q\) choose, in some way, a pair \((x_0,D_0)\in\mathscr B\) equivalent to it and a homeomorphism \(f:\{x_0\}\cup D_0\to\{y\}\cup D\). It is clear that \(fx=y\), for \(x\) and \(y\) are the only non-isolated points of the spaces \(\{x_0\}\cup D_0\) and \(\{y\}\cup D\). Hence \(f\) induces a certain one-to-one mapping \(\hat f:D_0\to D\).

The following is obvious.

Lemma 2. If for homeomorphisms \(f:\{x_0\}\cup D_0\to\{y\}\cup D\) and \(f':\{x_0\}\cup D_0\to\{y'\}\cup D\) the induced mappings \(\hat f\) and \(\hat f'\) coincide, then \(f=f'\) (i.e. \(y=y'\)).

Consider the families \(T=\{D:(x_0,D)\in\mathscr B\}\) and \(\Pi=\{D^*:D^*\subseteq A\}\). Lemma 2 means that the cardinality of the set \(Q\) is not greater than the cardinality of the set of all one-to-one mappings of the elements of the family \(T\) onto elements of the family \(Q\). It is clear that for any fixed \(D_0\in T\) and \(D_0^*\in\Pi\) there exist no more than \(c\) distinct one-to-one mappings of the set \(D_0\) onto the set \(D_0^*\) (for \(D_0\) and \(D_0^*\) are countable sets).

But the cardinality of \(T \leq c\) and the cardinality of \(\Pi \leq c\). Hence also the cardinality of the set of all possible pairs of elements of the families \(\Pi\) and \(T\) does not exceed \(c\). It follows that the cardinality of \(Q \leq c\). Proposition 3 is proved.

The set \([A]\setminus A\), as the remainder of the Čech extension of a countable discrete set, has cardinality \(2^c\). Denote by \(Q^1\) the set of first elements of the pairs of the family \(Q\). By Proposition 3, the cardinality of \(Q^1 \leq c\); therefore the set \(([A]\setminus A)\setminus Q^1\) is nonempty. We shall show that no point \(x^*\in ([A]\setminus A)\setminus Q^1\) can be carried to the point \(x_0\) by a homeomorphism of the space \(P\) onto the space \(P\).

Suppose the contrary; let \(g:P\to P\) be a homeomorphism of the space \(P\) onto the space \(P\) (possibly, onto a subspace \(P\)), with \(gx^*=x_0\). Since \(A\) is a discrete subspace of the space \(P\), the set \(gA\), which we denote by \(A_0\), is a discrete subspace of the space \(X\). We have \(x_0\in [A_0]\setminus A_0\), since \(gx^*=x_0\) and \(x^*\in [A]\setminus A\). Thus \((x_0,A_0)\), \((x^*,A)\) are pairs, and \((x_0,A_0)\in \mathscr F\), and \((x_0,A_0)\approx (x^*,A)\). By the definition of the family \(\mathscr E\), the pair \((x_0,A_0)\) is weakly equivalent to some pair belonging to the family \(\mathscr E\). This means that some subpair \((x_0,D_0)\) of the pair \((x_0,A_0)\) is equivalent to some pair belonging to the family \(\mathscr B\). From \((x_0,A_0)\approx (x^*,A)\) it follows, obviously, that the subpair \((x_0,D_0)\) of the pair \((x_0,A_0)\) is equivalent to some subpair \((x^*,D)\) of the pair \((x^*,A)\). Since the relation \(\approx\) is transitive, we can conclude that the pair \((x^*,D)\) is equivalent to some pair belonging to the family \(\mathscr B\). But then \((x^*,D)\in Q\)—contrary to the fact that \(x^*\notin Q^1\).

Definition 1. A subspace \(Y\) of a space \(X\) is called homogeneous in \(X\) if for any two points \(x,y\) of \(Y\) there exists a homeomorphism \(f\) from the space \(Y\) into \(X\) such that \(fx=y\).

Above it was proved

Main Proposition 4. An infinite bicompact subspace \(P\) of an extremally disconnected separable space \(X\) is not homogeneous in \(X\).

Here is an extremely useful obvious assertion, distinguishing the concept of “homogeneity in” from ordinary homogeneity.

Proposition 5. If \(X\supset Y\supset Z\) and \(Y\) is homogeneous in \(X\), then \(Z\) is homogeneous in \(X\).

From Main Proposition 4 and Proposition 5 it follows immediately that

Main Theorem 1. If \(Y\) is a subspace homogeneous in \(X\) of a separable extremally disconnected space \(X\), then the \(k\)-leader of the space \(Y\) is discrete (see (2)).

Corollary 1. A separable extremally disconnected homogeneous \(k\)-space is discrete and countable.

Since the preimage of a separable \(k\)-space under an irreducible perfect mapping is a separable \(k\)-space (see (2)), Theorem 1 implies

Corollary 2. The absolute (see (3)) of a nondiscrete separable \(k\)-space is nonhomogeneous.

Corollary 3. The absolute of a separable nondiscrete \(p\)-space (see (4)) is nonhomogeneous.

Corollary 4. The absolute of a separable nonmetrizable space is nonhomogeneous.

Here by a nonmetrizable space is meant any space not representable as the sum of a countable family of its locally zero-dimensional (in the sense of ind) subspaces. In this case the natural mapping of the absolute onto the space cannot be finite-to-one. Now Corollary 2 from Proposition 1 and Propositions 4 and 5 are applied.

Theorem 2. If \(Y\) is a subspace homogeneous in \(X\) of an extremally disconnected space \(X\) of weight \(c\), then the \(k\)-leader (see (2)) of the space \(Y\) is discrete.

Lemma 3. Every extremally disconnected space \(X\) of weight \(\tau\) is embedded in the absolute of the space \(D^\tau\).

We may assume that \(X\) lies in \(D^\tau\). Therefore it is enough to prove the following Lemma 4.

Lemma 4. If \(X \subset \widetilde X\), where \(X\) is extremally disconnected, and a perfect mapping \(f: Z \to \widetilde X\) is given, then the space \(X\) is homeomorphic to some subspace of the space \(Z\).

Indeed, consider \(Z_1 = f^{-1}X\) and \(f_1 = f|Z_1\), \(f_1: Z_1 \to X\). The mapping \(f_1\) is perfect; therefore there exists a closed set \(Z_2\) in \(Z_1\) such that \(fZ_2 = X\) and the mapping \(f_2 = f|Z_2\) is irreducible. Since \(X\) is extremally disconnected, \(Z_2\) is homeomorphic to \(X\)—Lemma 2 is proved.

We now prove Theorem 2. Applying Lemmas 3 and 4, we conclude that \(X\) is homeomorphic to a subspace of the absolute of the space \(D^c\). But \(D^c\), and hence its absolute, is separable. Now \(Y\) is a subspace of an extremally disconnected separable space. We can apply Theorem 1. Theorem 2 is proved.

Corollary 1. An extremally disconnected homogeneous \(k\)-space of weight \(c\) is discrete.

Corollary 2. An infinite extremally disconnected bicompactum of weight \(c\) is nonhomogeneous. Any infinite bicompactum containing one is also nonhomogeneous.

Remark. Since \(\beta N\) is the absolute of the simplest infinite compactum, the nonhomogeneity of the remainder \(\beta N \setminus N\) is a simple special case of Theorem 1. The first proofs of the nonhomogeneity of \(\beta N \setminus N\) were based on the continuum hypothesis (see, in particular, (5)); later Z. Frolík showed how to dispense with the latter.

The answers to the following questions are unknown to the author:

1°. Does there exist an infinite homogeneous extremally disconnected bicompactum?

2°. Can a nondiscrete extremally disconnected space be a group space?

3°. Is it true that the absolute of an arbitrary bicompactum of weight \(c\) is homogeneous?

4°. For which spaces \(X\) is the supremum of the cardinalities of the closures of discrete subspaces of the space \(X\) equal to the cardinality of \(X\)?

Moscow State University
named after M. V. Lomonosov

Received
5 X 1966

REFERENCES

1. M. H. Stone, Trans. Am. Math. Soc., 41, 375 (1937).
2. A. V. Arhangel’skii, Tr. Mosk. matem. obshch., 13, 3 (1965).
3. V. I. Ponomarev, DAN, 149, 26 (1963).
4. A. V. Arhangel’skii, Matem. sborn., 67 (109), 1, 55 (1965).
5. L. Gillman, M. Jerison, Rings of Continuous Functions, Princeton, 1960.

* This fact was discovered by V. I. Ponomarev and me in a friendly conversation.