UDC 513.83

MATHEMATICS

A. G. EL’KIN

ON INDECOMPOSABLE SPACES AND TOPOLOGICALLY SELF-DENSE ULTRAFILTERS

(Presented by Academician P. S. Aleksandrov, 21 IV 1969)

The basic notion in this note is that of a \(d\)-end of a topological space (Definition 1). With its help, in the first part we obtain a generalization of Šoke’s theorem on topologically self-dense ultrafilters (Theorem 1), and in the second—the existence of connected \(H\)-closed \(k\)-decomposable, \(1 \leq k < \aleph_0\), but not \((k+1)\)-decomposable spaces (Theorem 7), and the coabsoluteness of an arbitrary self-dense Hausdorff space with a minimal space (Theorem 9). We also note Theorem 2, which makes it possible to obtain minimal relaxations of spaces by means of centered systems, and Corollary 2: a \(\theta\)-homeomorphism preserves extremal disconnectedness. Finally, in the third part the space of \(d\)-ends of a given topological space \(X\) is constructed and its relation to the space \(\theta(X)\) of all ends of the space \(X\) is clarified.

Definition 1. A system of subsets of a topological space \(X\) is called \(d\)-centered* if the intersection of any finite family of its elements is dense in \(X\).

A \(d\)-centered system of subsets of the space \(X\) is called a \(d\)-end of the space \(X\) if it is not contained in any distinct \(d\)-centered system of subsets of \(X\). We shall denote the set of all \(d\)-ends of the space \(X\) by \(dX\).

It is easily proved that every \(d\)-centered system of subsets of the space \(X\) is contained in at least one \(d\)-end of the space \(X\). Put
\[ d_M=\{A\cap M:\ A\in d\}. \]

Proposition 1. \(1^\circ.\) Let \(d\in dX\) and \(M\subseteq X\). Then there exist disjoint canonical open subsets \(U\) and \(G\) in \(X\) such that
\[ [U\cup G]=X,\quad M\cap U\in d_U, \]
and there exists \(A\in d\) for which
\[ A\cap M\cap G=\Lambda. \]

\(2^\circ.\) Let \(d\) and \(d'\) be two \(d\)-ends of the space \(X\). Then there exist disjoint canonical open subsets \(U\) and \(G\) in \(X\) such that
\[ [U\cup G]=X,\quad d_U=d'_U, \]
and there exist \(A\in d\) and \(A'\in d'\) for which
\[ A\cap A'\cap G=\Lambda. \]

I. Šoke \((^{14})\) introduced the notion of a topologically self-dense ultrafilter (this is an ultrafilter in the set of points of the segment \([0,1]\) with a base of self-dense sets) and, under the assumption of the continuum hypothesis, proved the existence of such an ultrafilter. Below we give three proofs of the existence of ultrafilters with a base of self-dense sets in every self-dense \(T_0\)-space, not relying on the continuum hypothesis.

Lemma 1. Let** \(p\in\theta(X)\) and \(d\in dX\), where \(X\) is a nonempty topological space. Then
\[ p\wedge d=\{U\cap A:\ U\in p,\ A\in d\} \]
is a base of an ultrafilter in the set \(X\).

* Such systems were considered in \((^{3})\), p. 121.

** An end of the space \(X\) is a maximal centered system of open subsets in \(X\). The space \(\theta(X)\) is the set of all ends of the space \(X\), endowed with the topology whose base is formed by the sets
\[ O_U=\{p\in\theta(X):\ p\ni U\}. \]
In this topology \(\theta(X)\), as shown by S. Iliadis and S. V. Fomin \((^{7})\), is an extremally disconnected bicompactum.

Put \(tX=\min\{sU: U\ne \Lambda\ \text{and is open in }X\}\)*. From Lemma 1 it easily follows

Theorem 1. In every nonempty crowded \(T_0\)-space \(X\) \((tX\ne 1)\) there exists an ultrafilter with a base each element of which is a crowded subset of \(X\) of cardinality \(tX\), whose closure is a canonical closed set in \(X\).

Hence, by virtue of the well-known theorem of P. S. Aleksandrov stating that every zero-dimensional perfect compactum (i.e., discontinuum) is homeomorphic to the Cantor perfect set \(\mathcal C\), we immediately obtain

Corollary 1. There exists an ultrafilter in \(\mathcal C\) (and, consequently, in \([0,1]\)) with a base each element of which is a crowded countable set whose closure is homeomorphic to \(\mathcal C\).

This is the first half of Słupecki’s theorem \(((^{14})\), Theorem 10), proved by him under the assumption of the continuum hypothesis.

Second proof. We shall call a system of subsets of a space \(X\) \(q\)-centered if the intersection of every nonempty finite family of its elements is a nonempty crowded subset of \(X\). A maximal \(q\)-centered system of subsets of \(X\) will be called a \(q\)-end of the space \(X\). A space \(Y\) will be called a \(q\)-relaxation of the space \(X\) if \(Y\) is the set \(X\), endowed with the topology with prebase \(\mathcal T\cup q\), where \(\mathcal T\) is the topology of the space \(X\) and \(q\) is a \(q\)-centered system of subsets of \(X\).

Theorem 2. Let \(X\) be a crowded space and let \(q\) be a \(q\)-end in \(X\). Then the \(q\)-relaxation of the space \(X\) is a minimal space in which \(q\) is an end.

Since a minimal \(T_0\)-space is*** \(SI\) and since every end in a nonempty \(SI\), as shown in \((^5)\), forms a base of an ultrafilter, it follows from Theorem 2 that

Theorem 3. Every \(q\)-end of a nonempty crowded \(T_0\)-space \(X\) is a base of an ultrafilter in the set \(X\).

The third proof is simply a generalization of the second: let \(X\) be a nonempty crowded \(T_0\)-space; let \(Y\) be a minimal relaxation of the space \(X\), whose existence was proved by Hewitt \((^{13})\) and Katětov \((^8)\); then, as was already noted above, every end in \(Y\) is a base of an ultrafilter in the set \(Y\) and, consequently, in the set \(X\), and every set from this end is, obviously, crowded in the space \(X\).

II. Definition 2. A space \(Y\) is called a \(d\)-relaxation of the space \(X\) if \(Y\) is the set \(X\), endowed with the topology with prebase \(\mathcal T\cup d\), where \(\mathcal T\) is the topology of the space \(X\) and \(d\) is a \(d\)-centered system of subsets of \(X\).

Theorem 4. Let \(Y\) be a relaxation of the space \(X\). Then the following assertions are equivalent: (1) \(X\) and \(Y\) have one and the same supply of canonical open sets, or, equivalently, one and the same supply of canonical closed sets. (2) \(X\) and \(Y\) have one and the same associated semiregular space**. (3) \(Y\) is a \(d\)-relaxa-

* \(sX=\min\{|A|: A\ \text{is crowded in }X\}\).

** A space \(Y\) is called a relaxation of the space \(X\) if \(Y\) consists of the same points as \(X\), and if the identity mapping of \(Y\) onto \(X\) is continuous. A space is called minimal if it is crowded, and every one of its proper relaxations contains isolated points.

*** One says that \(X\) is \(SI\) if \(X\) is crowded and if every nonempty crowded subset of \(X\) is irresolvable, i.e., does not contain two disjoint dense subsets.

**** The canonical open subsets of the space \(X\) form a base of a semiregular topology on the set \(X\). The set \(X\), endowed with this topology, is called the semiregular space associated with \(X\) (see \((^3)\), p. 121).

spaces \(X\). (4) The identity mapping \(X\) onto \(Y\) is \(\theta\)-discontinuous*.

From Theorem 4 a number of important properties of \(d\)-relaxations follow easily.

Proposition 2. A \(d\)-relaxation of a connected space is connected.

Proposition 3. Every \(d\)-relaxation of an extremally disconnected space is extremally disconnected. Conversely, if a space has at least one extremally disconnected \(d\)-relaxation, then the space itself is extremally disconnected.

Proposition 4. A \(d\)-relaxation of an \(H\)-closed space is \(H\)-closed.

The last proposition follows from the fact that, as was proved by S. V. Fomin \((^{12})\), the \(\theta\)-continuous image of an \(H\)-closed space is \(H\)-closed.

Proposition 5. Every proper \(d\)-relaxation of an arbitrary topological space is non-semiregular (and hence irregular).

Lemma 2. A \(\theta\)-homeomorphism between two semiregular spaces is a homeomorphism.

Theorem 5. The following assertions concerning topological spaces \(X\) and \(Y\) consisting of the same points are equivalent:
(1) \(X\) and \(Y\) have one and the same stock of canonical open sets.
(2) \(X\) and \(Y\) have one and the same associated semiregular space.
(3) \(X\) and \(Y\) are \(d\)-relaxations of one and the same (semiregular) space.
(4) \(X\) and \(Y\) are identically \(\theta\)-homeomorphic.

Corollary 2. If two spaces are \(\theta\)-homeomorphic and one of them is extremally disconnected, then the other space is also extremally disconnected.

Theorem 6. Let \(X\) be a dense-in-itself \(T_1\)-space and \(d \in dX\). Then the \(d\)-relaxation of the space \(X\) is an \(MI\)-space**.

At the seminar of P. S. Aleksandrov, B. A. Efimov posed the following problem: do there exist connected Hausdorff \(k\)-decomposable*** spaces, \(2 \le k < \aleph_0\), which are not \((k+1)\)-decomposable? Its solution is given by

Theorem 7. Every connected \(H\)-closed \(k\)-decomposable space, \(1 \le k < \aleph_0\), has a connected \(H\)-closed \(k\)-decomposable relaxation that is not \((k+1)\)-decomposable****.

Theorem 8. Every dense-in-itself extremally disconnected \(H\)-closed space has an \(H\)-closed minimal relaxation*.

Theorem 9. Every dense-in-itself Hausdorff space is coabsolute** with some minimal space.

* A mapping \(f:X \to Y\) is called \(\theta\)-continuous in the sense of S. V. Fomin \((^{12})\) if, for every point \(x \in X\) and for every neighborhood \(Ofx\) of its image, there exists a neighborhood \(Ox\) of this point such that \(f[Ox] \subseteq [Ofx]\). A one-to-one mapping that is \(\theta\)-continuous in both directions is called a \(\theta\)-homeomorphism.

** A space is called an \(MI\)-space if it is dense in itself and if every subset dense in it is open.

*** A space is \(k\)-decomposable if it contains \(k\) pairwise disjoint subsets dense in it \((k \ge 1)\).

**** Hewitt’s problem \((^{13})\)—whether there exist connected Hausdorff indecomposable spaces—was solved by Padmavally \((^9)\) by the construction of an example. Then Bourbaki \(((^3), p. 181)\) and Anderson \((^2)\) gave a proof of the existence of a connected indecomposable relaxation for every connected Hausdorff space. The existence of \(k\)-decomposable Hausdorff spaces, \(2 \le k < \aleph_0\), which are not \((k+1)\)-decomposable, was proved in \((^4)\).

***** Dense-in-itself extremally disconnected \(H\)-closed spaces exist: for every dense-in-itself Hausdorff space \(X\), such a space is \(\theta(X)\). The existence of \(H\)-closed minimal spaces was proved by Katětov \((^8)\).

** Two Hausdorff spaces are coabsolute if they have one and the same absolute, i.e. the maximal irreducible \(\theta\)-continuous preimage. For the first time, the (spectral) theory of absolutes of paracompact Hausdorff spaces was constructed by V. I. Ponomarev \((^{10})\). Then, by the method of centered systems, S. Iliadis \((^6)\) and, by the spectral method, V. I. Ponomarev \((^{11})\) constructed it already for arbitrary Hausdorff spaces.

III. Definition 3. Let \(X\) be an arbitrary topological space. The space \(dX\) is the set \(dX\), endowed with the topology whose base is formed by the sets
\[ O_A=\{d\in dX:\ d\supset A\}, \]
where \(A\subset X\).

Proposition 6. For every space \(X\), the space \(dX\) is zero-dimensional and Hausdorff (and hence completely regular).

Theorem 10. Let \(\{U_\lambda:\lambda\in \mathscr L\}\) be a family of pairwise disjoint open subsets of the space \(X\), whose union is dense in \(X\). Then \(dX\) is homeomorphic to the product
\[ \prod\{dU_\lambda:\lambda\in \mathscr L\}, \]
endowed with the box topology.

Denote by \(R(X)\) the maximal open decomposable subspace of the space \(X\) (such exists).

Corollary 3. For every space \(X\), the space \(dX\) is homeomorphic to \(dR(X)\).

Proposition 7. If \(X\) is a nonempty decomposable Hausdorff space, then \(dX\) is not bicompact.

The construction, due to P. S. Aleksandrov \((^1)\), of the Stone–Čech extension of an arbitrary completely regular space as applied to a discrete space \(T\) gives the following: \(\beta T\) is the set of all ultrafilters in \(T\) with the topology whose base is formed by the sets
\[ O_E=\{\mathfrak F\in\beta T:\ \mathfrak F\supset E\}, \]
where \(E\subset T\). Thus, Lemma 1 defines a natural mapping
\[ \mu:\theta(X)\times dX\to \beta T, \]
where \(T\) is the set \(X\), endowed with the discrete topology. Namely: \(\mu(p,d)\) is the ultrafilter with base \(p\wedge d\).

Theorem 11. The mapping \(\mu\) is continuous and open, and for each \(d\in dX\) the mapping \(\mu\) on the set \(\theta(X)\times\{d\}\) is a homeomorphism.

Corollary 4 (B. A. Efimov)*. Every regular extremally disconnected space \(X\) is topologically contained in \(\beta T\), where \(T\) is a discrete space of cardinality
\[ s\beta X=s\theta(X)\le sX. \]

Corollary 5. Every infinite extremally disconnected bicompactum (in particular \(\beta N\), and in general \(\beta T\), where \(T\) is an infinite discrete space) contains an indecomposable space**.

In conclusion, the author expresses sincere gratitude to his scientific adviser V. I. Ponomarev for posing the problems and discussing the results.

Mechanical-Mathematical Faculty
of Moscow State University
named after M. V. Lomonosov

Received
16 IV 1969

REFERENCES

1. P. S. Aleksandrov, Matem. sborn., 5 (47), No. 2, 403 (1939).
2. D. R. Anderson, Proc. Am. Math. Soc., 16, No. 3, 463 (1965).
3. N. Bourbaki, General Topology, Basic Structures, Moscow, 1968.
4. A. G. El’kin, Vestn. Mosk. Univ., Ser. 1, Math., Mech., No. 4 (1969).
5. A. G. El’kin, ibid., Ser. 1, Math., Mech., No. 5 (1969).
6. S. Iliadis, DAN, 149, No. 1, 22 (1963).
7. S. Iliadis, S. Fomin, UMN, 21, issue 4 (1966).
8. M. Katetov, Matem. sborn., 21 (63), No. 1, 3 (1947).
9. K. R. Padmavally, Duke Math. J., 20, No. 4, 513 (1953).
10. V. I. Ponomarev, Matem. sborn., 60 (102), issue 1, 89 (1963).
11. V. I. Ponomarev, DAN, 149, No. 1, 26 (1963).
12. S. Fomin, Ann. Math., 44, No. 3, 471 (1943).
13. E. Hewitt, Duke Math. J., 10, 309 (1943).
14. G. Choquet, Bull. Sci. math., 2 ser., 92, No. 1–2, 41 (1968).

* Unpublished.

** A space is indecomposable if it is dense in itself and is not decomposable.