UDC 513.831

MATHEMATICS

A. V. ARKHANGEL'SKII

THE SUSLIN NUMBER AND CARDINALITY, CHARACTERS OF POINTS IN SEQUENTIAL BICOMPACTS

(Presented by Academician P. S. Aleksandrov on 20 X 1969)

The notation and definitions used here are explained in \((^1,^2,^{4-7})\). We note specifically the following. Of topological spaces, only Hausdorff spaces are considered; of cardinal numbers, only infinite ones. The latter are denoted by the symbols \(\lambda,\tau\). The character of a set \(A\) in a space \(X\) is \(\chi(A,X)=\min\{|B|: B\) is a base of \(A\) in \(X\}\), where a base of \(A\) in \(X\) is a family of open sets containing arbitrarily small neighborhoods of \(A\). If \(x\in X\), then \(\chi(x,X)=\chi(\{x\},X)\). The character of a space \(X\) is \(\chi(X)=\sup\{\chi(x,X):x\in X\}\)*. The combination (G.C.H.) means that the generalized continuum hypothesis is assumed to hold. By \(|\alpha|\) is denoted the cardinality of the set well ordered by type \(\alpha\). If \(M\) is a set, then \(\exp M=\{A:A\subset M\}\) and \(\exp_\tau M=\{A:A\subset M\text{ and }|A|\le \tau\}\).

Definition 1. The height \(h(x,X)\) of a space \(X\) at a point \(x\) is
\[ \min\{\chi(F,X):x\in F\subset X,\ F\text{ is bicompact}\}. \]

Definition 2. The height \(h(X)\) of a space \(X\) is \(\sup\{h(x,X):x\in X\}\).

Definition 3. The bitightness \(bt(X)\) of a space \(X\) is the least of those \(\tau\) such that if \(M\subset X\) and \([M]\ne M\), then there exist \(x\in X\setminus M\) and a family \(\lambda\subset \exp_\tau M\) for which: a) \(|\lambda|\le \tau\) and b) \(\{x\}=\cap\{[P]:P\in\lambda\}\).

Definition 4. The divergence \(\operatorname{div}(X)\) of a space \(X\) is the least \(\tau\) such that for every set \(M\) not closed in \(X\) there exist \(x\in [M]\setminus M\) and a family \(\xi\subset \exp_\tau M\) converging** to \(x\), for which \(|\xi|\le\tau\).

Proposition 1. \(t(X)\le bt(X)\le \operatorname{div}(X)\le \chi(X)\)***.

Proposition 2. If \(Y\) is a quotient space of a space \(X\), then \(\operatorname{div}(Y)\le \operatorname{div}(X)\).

Lemma 1. If \(A\subset X\), then \(|[A]|\le |A|^{bt(X)}\).

Proof. We shall assume that \(A\) is infinite. Put \(bt(X)=\tau\). Define, by transfinite induction, for every ordinal number \(\alpha\) less than \(\tau^+\), the set \(A_\alpha\) as follows: \(A_0=A\); if \(A_\alpha\) is defined, then
\[ A_{\alpha+1}=\{x\in X:\text{ there exists }\lambda\in \exp_\tau(\exp_\tau A),\text{ for which }\{x\}=\cap\{[P]:P\in\lambda\}\}; \]
if \(\beta\) is a limit ordinal and the sets \(A_\alpha\) are defined for all \(\alpha<\beta\), then \(A_\beta=\cup\{A_\alpha:\alpha<\beta\}\). We shall show that: I. \(|A_\alpha|\le |A|^\tau\) for all \(\alpha<\tau^+\), and II. \(\cup\{A_\alpha:\alpha<\tau^+\}\) is a closed set.

Suppose that relation I is not true, and let \(\alpha_0\) be the first transfinite ordinal for which it is violated. Then \(\alpha_0>0\), since \(|A_0|=|A|\le |A|^\tau\). The transfinite ordinal \(\alpha_0\) cannot be a limit ordinal—in the contrary case \(|A_{\alpha_0}|\le\)

* If \(M\) is a set of cardinal numbers, then \(\sup M\) is the least of the cardinal numbers not smaller than every element of \(M\).

** A family \(\xi\) converges to a point \(x\) if every neighborhood of the point \(x\) contains some element of the family \(\xi\).

*** It is easily proved that the weak tightness \(t_c(X)\) and the tightness \(t(X)\) in the sense of \((^2)\) always coincide. Therefore \(t(X)=t_c(X)\le bt(X)\).

\(\leqslant \sum\{|A_\alpha|:\alpha<a_0\}\leqslant |A|^\tau\), since \(|a_0|<\tau^+\leqslant |A|^\tau\) and \(|A_\alpha|\leqslant |A|^\tau\) for all \(\alpha<a_0\) by the choice of \(a_0\). Hence, \(a_0=\alpha'+1\). But \(|A_{\alpha'}|\leqslant |A|^\tau\). It is known that for any set \(B\) the cardinality of the family \(\exp_\tau B\) does not exceed \(|B|^\tau\) \((^9)\). To each point \(x\in A_{\alpha'+1}\) assign some family \(\lambda(x)\) of subsets of the set \(A_{\alpha'}\), for which \(|\lambda(x)|\leqslant \tau\), \(|\bigcup\{P:P\in\lambda(x)\}|\leqslant \tau\), and \(\{x\}=\bigcap\{[P]:P\in\lambda(x)\}\). Obviously, this defines a one-to-one mapping of the set \(A_{\alpha'+1}\) into the set \(\exp_\tau(\exp_\tau A_{\alpha'})\). Hence,
\[ |A_{\alpha'+1}|\leqslant |\exp_\tau(\exp_\tau A_{\alpha'})|\leqslant (|A_{\alpha'}|^\tau)^\tau\leqslant |A|^\tau. \]
Relation I is proved.

Now suppose that relation II is not true, i.e., that the set \(C=\bigcup\{A_\alpha:\alpha<\tau^+\}\) is not closed. Then there exist a point \(z\notin C\) and a family \(\mu\in\exp_\tau(\exp_\tau C)\) such that \(\{z\}=\bigcap\{[P]:P\in\mu\}\). Obviously, for some \(a^*<\tau^+\), then \(\bigcup\{P,P\in\mu\}\subset\bigcup\{A_\alpha:\alpha<a^*\}\subset A_{a^*}\). From the definition of the set \(A_{a^*+1}\) it now follows that \(z\in A_{a^*+1}\), a contradiction. Thus, \(C\) is a closed set. Moreover, by I, \(|C|\leqslant \tau^+|A|^\tau\leqslant |A|^\tau\). But \(A=A_0\subset C\); consequently, \([A]\subset C\) and \(|[A]|\leqslant |A|^\tau\). Lemma 1 is proved.

We shall need the following two assertions:

VI (transitivity of character). If \(X_2\subset X_1\subset X\), where \(X_2\) and \(X_1\) are bicompacts, then \(\chi(X_2,X)\leqslant \chi(X_2,X_1)+\chi(X_1,X)\) \((^3)\).

VII. If \(X\) is \(\tau\)-compact, \(bt(X)\leqslant \tau\), and \(\chi(X)\leqslant 2^\tau\), then \(|X|\leqslant 2^\tau\).

Assertion VII is obtained in an obvious way from the main theorem 2 of \((^1)\) by means of Lemma 1.

Lemma 2 (main). Let \(X\) be a bicompact, \(t(x,X)<\tau\leqslant \lambda\) for every point \(x\in X\), and let the cardinal number \(\tau\) be regular. Suppose further that from \(A\subset X\) and \(|A|<\tau\) it follows, for every \(A\subset X\), that \(|[A]|<\lambda\). Then there exists a closed set \(B\) in \(X\) for which \(\chi(B,X)<\tau\) and \(0<|B|<\lambda\).

Lemma 3. Every nonempty open subset of a regular space contains a nonempty closed-in-this-space set of type \(G_\delta\).

Proof of Lemma 2. If \(|X|<\lambda\), the assertion is obvious. If \(|X|\geqslant\lambda\), one can carry out a construction by transfinite induction, as a result of which to each \(\alpha<\tau\) there will be assigned a point \(x(\alpha)\) so that the set \(\{x(\alpha):\alpha<\tau\}\), well ordered in accordance with the order of the transfinite numbers, will be a free sequence of length \(\tau\) \((^1)\). In parallel, nonempty closed sets \(F(\alpha)\) will be defined.

As \(x(0)\) choose any point of the set \(X\). Take as \(F(0)\) some nonempty set closed in \(X\), of type \(G_\delta\), not containing the point \(x(0)\). Let \(\alpha_0<\tau\), and suppose that for every \(\alpha<\alpha_0\) a point \(x(\alpha)\) and a nonempty closed set \(F(\alpha)\subset X\) have already been defined so that \(\chi(F(\alpha),X)\leqslant |\alpha|+\aleph_0\), and if \(\alpha_1<\alpha_2<\alpha_0\), then \(F(\alpha_1)\supset F(\alpha_2)\). Introduce the notation \(A(\alpha_0)=\{x(\alpha):\alpha<\alpha_0\}\). Then \(|A(\alpha_0)|\leqslant |\{\alpha:\alpha<\alpha_0\}|<\tau\), and therefore \(|[A(\alpha_0)]|<\lambda\). Put \(B(\alpha_0)=\bigcap\{F(\alpha):\alpha<\alpha_0\}\). If \(|B(\alpha_0)|<\lambda\), then \(B(\alpha_0)\) is the desired set, for \(\{F(\alpha):\alpha<\alpha_0\}\) is a centered family of nonempty, closed-in-\(X\) sets, \(X\) is a bicompact, the character and pseudocharacter of a closed set in a bicompact coincide, and \(\sum\{|\alpha|:\alpha<\alpha_0\}\leqslant |\alpha_0|+\aleph_0<\tau\). If \(|B(\alpha_0)|\geqslant\lambda\), then \(B(\alpha_0)\setminus [A(\alpha_0)]\ne \Lambda\). Take as \(x(\alpha_0)\) any point of the set \(B(\alpha_0)\setminus [A(\alpha_0)]\). Define \(F(\alpha_0)\) as some nonempty set, closed in \(X\), contained in \(B(\alpha_0)\setminus([A(\alpha_0)]\cup\{x(\alpha_0)\})\), of type \(G_\delta\) in \(B(\alpha_0)\). By VI,
\[ \chi(F(\alpha_0),X)\leqslant \chi(B(\alpha_0),X)+\aleph_0\leqslant |\alpha_0|+\aleph_0. \]

We shall show that the described process of defining the points \(x(\alpha)\) and the sets \(F(\alpha)\), under any realization of it, stops at some transfinite ordinal smaller than \(\tau\). The realization determines only which particular transfinite ordinal \(\alpha^*<\tau\) turns out to be obstructing. For this \(\alpha^*\) we then have \(|B(\alpha^*)|<\lambda\) and \(\chi(B(\alpha^*),X)\leqslant |\alpha|+\aleph_0<\tau\), whence it follows that \(B(\alpha^*)\) is the desired set.

…set. Suppose, however, that the sets \(F(\alpha)\) and the points \(x(\alpha)\) have been defined for all \(\alpha<\tau\). Put \(A=\{x(\alpha): \alpha<\tau\}\), and \(x(\alpha_1)<x(\alpha_2)\) if and only if \(\alpha_1<\alpha_2\). For \(\alpha_1\ne\alpha_2\), \(x(\alpha_1)\ne x(\alpha_2)\). Hence \(|A|=\tau\). By the construction, if \(\alpha_0<\tau\), then \([\{x(\alpha):\alpha\le\alpha_0\}]\cap F(\alpha_0)=\Lambda\), and if \(\beta>\alpha_0\), then \(x(\beta)\in F(\alpha_0)\). Therefore
\[ [\{x(\alpha):\alpha\le\alpha_0\}]\cap\{x(\beta):\alpha_0<\beta\}\subset [\{x(\alpha):\alpha\le\alpha_0\}]\cap F(\alpha_0)=\Lambda. \]
Thus it has been proved that \(A,<\) is a free sequence of length \(\tau\). But then some point \(y\in X\) is a point of complete accumulation for \(A\). Now a simple fact is needed (see \((^1)\)).

Lemma 4. If the tightness at every point of a topological space \(X\) is less than \(\tau\), and \(\tau\) is a regular cardinal number, then in \(X\) there is no point of complete accumulation for any free sequence of length \(\tau\).

Lemma 5. Suppose that all the assumptions of Lemma 2 are satisfied. Then there exists a family \(\gamma\) of pairwise disjoint closed subsets of \(X\) such that \([\bigcup\{P:P\in\gamma\}]=X\), and for every \(P\in\gamma\), \(\chi(P,X)<\tau\) and \(0<|P|<\lambda\).

Lemma 5 is easily proved on the basis of the Kuratowski–Zorn principle.

Definition 5. A family \(\mathcal E=\{\gamma_\alpha:\alpha\in M\}\) of families of subsets of a set \(X\) is called Hausdorff if, for any \(\alpha_1,\alpha_2\in M\), where \(\alpha_1\ne\alpha_2\), there exist \(P_1\in\gamma_{\alpha_1}\) and \(P_2\in\gamma_{\alpha_2}\) such that \(P_1\cap P_2=\Lambda\). A family \(\gamma\) of sets is called a prefilter if from \(P'\in\gamma\), \(P''\in\gamma\) it follows that there exists \(P\in\gamma\) for which \(P\subset P'\cap P''\).

Lemma 6. Let \(\mathcal E=\{\gamma_\alpha:\alpha\in M\}\) be a Hausdorff family of prefilters in \(X\), with \(|M|>2^\tau\) and \(|\gamma_\alpha|\le\tau\) for every \(\alpha\in M\). Then there exist a set \(M'\subset M\) and sets \(P_\alpha\in\gamma_\alpha\) for every \(\alpha\in M'\), such that \(|M'|>\tau\) and the family \(\{P_\alpha:\alpha\in M'\}\) is disjoint.

Lemma 6 easily follows from Lemma 6 of paper \((^4)\). Lemmas 7 and 8 are obvious.

Lemma 7. If \(x\in U\subset X\), where \(U\) is open in \(X\), and \(h(x,X)\le\tau\), then there exists a bicompact \(\Phi\subset X\) for which \(x\in\Phi\subset U\) and \(\chi(\Phi,X)\le\tau\).

Lemma 8. If \(h(X)\le\tau\), then there exists a disjoint family \(S\) of bicompact subsets of the space \(X\) such that \((j_1):[\bigcup\{F:F\in S\}]=X\), and \((j_2):\chi(F,X)\le\tau\) for every \(F\in S\).

Lemma 9. If \(X\) is bicompact and \(bt(X)\le\tau\), then there exists a disjoint family \(\gamma\) of closed subsets of \(X\) such that: \((k_1):[\bigcup\{P:P\in\gamma\}]=X\); \((k_2):\) if \(P\in\gamma\), then \(|P|\le 2^\tau\), and \((k_3):\chi(P,X)\le\tau\) for all \(P\in\gamma\).

Proof. All the assumptions of Lemma 2 are satisfied with respect to the space \(X\) and the cardinal numbers \(\tau^+\) in the role of \(\tau\) and \((2^\tau)^+\) in the role of \(\lambda\), since \(\tau^+\) is regular, \(t(X)\le bt(X)<\tau^+\), and if \(A\subset X\), \(|A|<\tau^+\), then \(|A|\le\tau\) and \(|[A]|\le\tau^\tau=2^\tau<(2^\tau)^+=\lambda\) by Lemma 1. Therefore Lemma 5 is applicable. Lemma 9 is proved.

Lemma 10. If \(F\) is closed in \(X\), then \(bt(F)\le bt(X)\).

Theorem 1. \(|X|\le 2^{bt(X)+c(X)+h(X)}\).

Proof. Choose a family \(S\) of nonempty subsets of the space \(X\) in accordance with Lemma 8. To each \(F\in S\) apply Lemma 9—in accordance with it choose a family \(\gamma_F\) of nonempty subsets of the space \(F\). Put \(\mathcal E=\{\gamma_F:F\in S\}\). By the transitivity of character \((\mathrm{VI})\), \(\chi(\Phi,X)\le bt(X)+h(X)\) for every \(\Phi\in\mathcal E\). To each \(\Phi\in\mathcal E\) assign some base \(\xi_\Phi\) of the set \(\Phi\) in \(X\), for which \(|\xi_\Phi|\le bt(X)+h(X)\). Since \(X\) is a Hausdorff space, and the family \(\mathcal E\) consists of pairwise disjoint nonempty bicompacts, \(\{\xi_\Phi:\Phi\in\mathcal E\}\) is a Hausdorff family of prefilters. If we had
\[ |\mathcal E|>2^{c(X)+bt(X)+h(X)}, \]
then, by Lemma 6, there would be a disjoint family of elements of these prefilters whose cardinality is greater than \(c(X)\), and this is impossible, since the prefilters \(\xi_\Phi\), \(\Phi\in\mathcal E\), consist of open sets. Hence
\[ |\mathcal E|\le 2^{c(X)+bt(X)+h(X)}. \]
From this and from condition \((k_2)\) of Lemma 9 it follows that
\[ |\bigcup\{\Phi:\Phi\in\mathcal E\}|\le 2^{c(X)+bt(X)+h(X)}. \]
But, obviously, \([\bigcup\{\Phi:\Phi\in\mathcal E\}]=X\). By Lemma 1 it follows that
\[ |[\bigcup\{\Phi:\Phi\in\mathcal E\}]| \le |\bigcup\{\Phi:\Phi\in\mathcal E\}|^{bt(X)} \le \left(2^{bt(X)+c(X)+h(X)}\right)^{bt(X)} = 2^{bt(X)+c(X)+h(X)}. \]

Theorem 1 is proved.

Theorem 2. \([\{x \in X : \chi(x,X) \leq 2^{bt(X)} + h(X)\}] = X.\)

Proof. We apply Lemmas 8 and 9, in particular using condition \((k_2)\) from the formulation of Lemma 9 (note that if \(F\) is compact, then \(\chi(x,F) \leq |F|\)); we shall further refer to the transitivity of character (VI).

Theorem 3 (G.C.H.). \([\{x \in X : \chi(x,X) \leq bt(X) + h(X)\}] = X.\)

Proof. Choose in \(X\) (Lemmas 8 and 9) a disjoint family \(\mathcal E\) of compacta whose character in \(X\) does not exceed \(h(X) + bt(X)\), and whose cardinality does not exceed \(2^{bt(X)}\), such that \([\bigcup\{\Phi:\Phi \in \mathcal E\}] = X\). It is known \((^8)\) that if \(X\) is a compactum and \(\chi(x,X) > \tau\) for all \(x \in X\), then \(|X| \geq 2^{(\tau+)}\). Assuming that \(2^{(\tau+)} > 2^\tau\) (this follows from the generalized continuum hypothesis), we conclude that in each \(\Phi \in \mathcal E\) the points whose character in \(\Phi\) does not exceed \(bt(X)\) form an everywhere dense subset. The conclusion of the theorem now follows from the transitivity of character.

A topological space is called homogeneous if for any \(x,y \in X\) there exists a homeomorphism \(f:X \to X\) such that \(f(x)=y\) and \(f(X)=X\).

Theorem 4. If \(X\) is homogeneous, then \(\chi(X) \leq 2^{bt(X)} + h(X)\).

Theorem 5 (G.C.H.). If \(X\) is homogeneous, then \(\chi(X) \leq bt(X) + h(X)\).

Theorem 6. If \(X\) is homogeneous, then \(|X| \leq 2^{ic(X)+bt(X)+h(X)}\) (see \((^1)\)).

Theorem 4 follows from Theorem 2, Theorem 5 follows from Theorem 3, and Theorem 6 follows from Theorem 4 and \((^1)\) (see Theorem 2).

1st group of corollaries. Let \(X\) be a space of point-countable type. Then:
a) if \(X\) is sequential and satisfies Suslin’s condition, then \(|X| \leq 2^{\aleph_0}\);
b) if \(X\) is sequential, then \([\{x \in X:\chi(x,X) \leq 2^{\aleph_0}\}] = X\);
c) (C.H.) if \(X\) is sequential, then \([\{x \in X:\chi(x,X) \leq \aleph_0\}] = X\);
d) \([\{x \in X:\chi(x,X) \leq 2^{bt(X)}\}] = X = [\{x \in X:\chi(x,X) \leq 2(2^{t(x)})\}]\);
e) (G.C.H.) \([\{x \in X:\chi(x,X) \leq bt(X)\}] = X = [\{x \in X:\chi(x,X) \leq 2^{t(X)}\}]\);
f) (C.H.) if \(X\) is a quotient space of a space with the first axiom of countability, then \([\{x \in X:\chi(x,X) \leq \aleph_0\}] = X\).

2nd group of corollaries. Let \(X\) be a homogeneous compactum. Then:
a) if the space \(X\) is sequential, then \(|X| \leq 2^{\aleph_0}\);
b) (C.H.) if the space \(X\) is sequential, then \(\chi(X) \leq \aleph_0\);
c) \(|X| \leq 2^{bt(X)}\);
d) (G.C.H.): \(\chi(X) \leq bt(X)\).

Note added in proof. The following holds.

Theorem 7 (G.C.H.). The cardinality of a homogeneous compactum cannot be a limit cardinal number.

From the proofs of Lemmas 2, 5 and Lemmas 8, 9 there evidently follows

Theorem 8. If \(h(X) \leq \aleph_0\) and \(c(X') \leq \aleph_0\) for all \(X' \subset X\), then
\(X = [X^*]\), where \(|X^*| \leq 2^{\aleph_0}\).

An analogue of this theorem is true for any \(\tau\).

The idea of formulating Theorem 8 was suggested to me by B. Shapirovskii after he had read an oral presentation of this work.

Mechanics and Mathematics Faculty
Moscow State University
named after M. V. Lomonosov

Received
15 X 1969

CITED LITERATURE

1. A. V. Arhangel’skii, DAN, 187, No. 5, 967 (1969).
2. A. V. Arhangel’skii, DAN, 184, No. 4, 767 (1969).
3. M. M. Choban, Vestn. Mosk. Univ., No. 6, 87 (1967).
4. I. Juhász, A. Hajnal, Proc. Koninkl. Nederl. Akad. Wet., Ser. A, 70, No. 3, 343 (1967).
5. A. V. Arhangel’skii, DAN, 181, No. 6, 1303 (1968).
6. B. A. Efimov, Tr. Mosk. matem. ob., 14, 211 (1965).
7. A. V. Arhangel’skii, ibid., 13, 3 (1965).
8. E. Čech, B. Pospíšil, Publ. Faculté Sci. Univ. Masaryk, Brno, 258, 1 (1938).
9. W. Sierpiński, Cardinal and Ordinal Numbers, Warszawa, 1965.