MATHEMATICS

A. HAINAL, I. JUHÁSZ

SOME RESULTS IN SET-THEORETIC TOPOLOGY

(Presented by Academician P. S. Aleksandrov on 1 IV 1966)

In this note we shall state our results without proofs. The methods of proof belong to the so-called combinatorial set theory and, first of all, to the theory of “partition relations” of Erdős and Rado (see, for example, (⁶)).

§ 1. In this section we deal with the cardinality of discrete subspaces of topological spaces. As is known, recently de Groot (¹), B. A. Efimov (²), and J. Isbell (³) considered this question and proved, independently of one another, similar results. We have succeeded in improving these results and proving others.

Theorem 1. Every Hausdorff space \(R\) of cardinality greater than \(\exp \exp \mathfrak m\) contains a discrete subspace of cardinality greater than \(\mathfrak m\). (Here \(\mathfrak m\) is an arbitrary infinite cardinal number.)

This theorem follows at once from the set-theoretic lemma 1.

Lemma 1. Let \(H\) be an arbitrary set, and suppose that to each element \(x \in H\) there is assigned some system \(S(x)\) of subsets of the set \(H\) such that: a) from \(U, V \in S(x)\) it follows that \(U \cap V \in S(x)\); b) if \(x, y \in H\) and \(x \ne y\), then there exist \(U \in S(x)\) and \(V \in S(y)\) for which \(U \cap V = \varnothing\).

Then from \(|H| > \exp \exp \mathfrak m\) there follows the existence of a subset \(D \subset H\) for which \(|D| > \mathfrak m\) and \([S(x) \cap D] \setminus \{x\} = \varnothing\) for \(x \in D\).

In (¹) de Groot posed the problem: does there exist in every \(T_2\)-space of density greater than \(\mathfrak m\) a discrete subspace of cardinality \(>\mathfrak m\)? In connection with this problem we proved the following result.

Theorem 2. In every \(T_2\)-space of density \(>\exp \mathfrak m\) there is a discrete subspace of cardinality \(>\mathfrak m\).

The following two theorems are devoted to the converse of theorem 2.

Let \(s(R)\) be the density and \(w(R)\) the weight of the space \(R\). As is known, \(w(R) \le \exp s(R)\) for regular \(R\). Hence one immediately obtains

Theorem 3. If a regular space \(R\) contains a discrete subspace of cardinality \(>\exp \mathfrak m\), then \(s(R) > \mathfrak m\).

However, for Hausdorff spaces the analogous result does not hold.

Theorem 4. Let \(\mathfrak m\) be an arbitrary infinite cardinal number. Then there exists a Hausdorff space \(R\), containing a discrete subspace of cardinality \(\exp \exp \mathfrak m\), for which \(s(R) = \mathfrak m\).

The next theorem is interesting because it concerns discrete subspaces of \(T_1\)-spaces. As is known, the pseudocharacter \(\psi(x, R)\) of a point \(x\) of a space \(R\) is the least cardinality of such a system of neighborhoods of the point \(x\) whose intersection is equal to \(\{x\}\).

Theorem 5. Let \(R\) be a \(T_1\)-space and let \(\mathfrak m\) be an infinite cardinal number. Suppose that \(\psi(x, R) \le \mathfrak m\) at every point \(x \in R\) and

* It is easy to see that from a positive solution of this problem of de Groot there would follow a positive solution of Suslin’s problem. However, the very latest investigations in axiomatic set theory show that Suslin’s problem is independent of the usual axioms of set theory. Thus, in a certain sense theorem 2 can no longer be improved.

\(\exp \mathfrak m<|R|\). Then \(R\) contains a discrete subspace of cardinality \(>\mathfrak m\).

Corollary. Let \(R\) be a \(T_1\)-space satisfying the first axiom of countability. If \(|R|>\exp \mathfrak m\), then \(R\) contains a discrete subspace of cardinality \(>\mathfrak m\).

De Groot also posed the following problem: does every Hausdorff space \(R\) with \(|R|=\aleph_\lambda\) contain a discrete subspace of cardinality \(\aleph_\lambda\), if \(\lambda\) is a limit ordinal and if one assumes the generalized continuum hypothesis? In connection with this problem the following theorems have been obtained.

Theorem 6. Let \(R\) be a Hausdorff space and \(|R|=\aleph_{\lambda+1}\), where \(\lambda\) is a limit number. If one assumes the generalized continuum hypothesis, then \(R\) contains a discrete subspace of cardinality \(\aleph_\lambda\).

Theorem 7. Let \(R\) be a Hausdorff space and \(|R|=\aleph_\lambda\), where \(\lambda\) is a limit number, cofinal with \(\omega\) (that is, \(\lambda\) is representable as a countable sum of ordinal numbers smaller than \(\lambda\)). Then, if one assumes the generalized continuum hypothesis, \(R\) contains a discrete subspace of cardinality \(\aleph_\lambda\).

§ 2. In (⁴) the following results are proved: a) If \(R\) is a Hausdorff space, \(\mathfrak m\) is a regular cardinal number and \(s(R)>\mathfrak m\), then \(R\) contains a subspace \(R'\) for which \(s(R')=\mathfrak m\). b) An analogous result holds for singular \(\mathfrak m\), however only under the assumption of the generalized continuum hypothesis. There the problem is also posed (Problem 4.2.1): can b) be freed from this hypothesis? The following theorems give partial solutions of this problem.

Theorem 8. Let \(R\) be a Hausdorff space and \(s(R)>\aleph_\lambda\), where \(\lambda\) is a limit number cofinal with \(\omega\). Then \(R\) contains a subspace \(R'\) for which \(s(R')=\aleph_\lambda\).

Theorem 9. Let \(R\) be a linearly ordered topological space (where the basis of the topology is given by open intervals). If \(\mathfrak m\) is an arbitrary cardinal number and \(s(R)>\mathfrak m\), then \(R\) contains a subspace \(R'\) for which \(s(R')=\mathfrak m\).

§ 3. The character \(\chi(x,R)\) of a point \(x\) of a space \(R\) is the least cardinality of fundamental systems of neighborhoods of the point \(x\).

Theorem 10. Let \(R\) be a Hausdorff space and \(\mathfrak m\) an infinite cardinal number. If the set of those points \(x\in R\) for which \(\chi(x,R)\leq \mathfrak m\) has cardinality greater than \(\exp \mathfrak m\), then \(R\) contains more than \(\mathfrak m\) pairwise disjoint open subsets.

Let \(N\) be a countable discrete space and let \(\beta N\) be its Stone–Čech compactification. As E. Čech showed (⁵), for points \(x\in \beta N\setminus N\),

\[ \aleph_0<\chi(x,\beta N)\leq 2^{\aleph_0}. \]

One of the authors posed the problem: how can one estimate more precisely the characters of the points \(x\in \beta N\setminus N\)? The first result in this direction is Theorem 11, whose proof is easily obtained by using Theorem 10.

Theorem 11. The set of those points \(x\in \beta N\setminus N\) for which

\[ \exp[\chi(x,\beta N)]<\exp\exp\aleph_0 \]

has cardinality less than \(\exp\exp\aleph_0\). (It is known that \(|\beta N|=\exp\exp\aleph_0\).)

This result may be reformulated as follows: for almost all \(x\in \beta N\setminus N\) the equality

\[ \exp[\chi(x,\beta N)]=\exp\exp\aleph_0 \]

holds.

It seems to us that, independently of the continuum hypothesis, it is impossible to characterize precisely the characters of the points \(x\in \beta N\setminus N\).

L. Eötvös University,
Budapest, Hungary

Received
22 III 1966

REFERENCES

¹ J. de Groot, Bull. Acad. polon. Sci., 13, 537 (1965).
² B. A. Efimov, DAN, 164, 967 (1965).
³ I. R. Isbell, Czechoslovak Mathematical Journal, 14, 89 (1964).
⁴ I. Juhász, Ann. Univ. Sci. Budapest, Sectio Math., 8, 75 (1965).
⁵ E. Čech, Ann. Math., 38, 823 (1937).
⁶ P. Erdős, A. Hajnal, R. Rado, Acta Math. Acad. Sci. Hung., 16, 91 (1965).