UDC 519.52

MATHEMATICS

B. EFIMOV

ON THE CARDINALITY OF HAUSDORFF SPACES

(Presented by Academician P. S. Aleksandrov on 26 II 1965)

In this note it is proved that, with the increase of the cardinality of a Hausdorff space \(X\), the upper bound of the cardinalities of its discrete, separated, and dispersed subspaces increases. The order of growth is established for each of these three types. For discrete spaces the order of growth is not less than \(\log\log\log |X|\), for separated spaces \(\log\log |X|\), and for dispersed spaces \(\log |X|\), where \(|X|\) denotes the cardinality of the set \(X\), and \(\log n\) is the least cardinal number \(m\) for which \(\exp m = 2^m \ge n\). Hence it follows, in particular, that the cardinality of every Hausdorff space satisfying Suslin’s condition* hereditarily does not exceed \(\exp \exp \exp \aleph_0\). We note that there exist bicompacts of arbitrarily high cardinality satisfying Suslin’s condition, for example dyadic ones \((^2)\). On the other hand, for dyadic bicompacts the hereditary fulfillment of Suslin’s condition is equivalent to their metrizability \((^3)\).

Let \(m\) be an infinite cardinal number, and \(j\) a natural number. Put \(\varphi_j(m)=\exp\ldots\exp \aleph_\rho\), if \(m\) is a number of the form \(\aleph_{\rho+1}\), or

\[ \varphi_j(m)=\sum_{\alpha\in A}\underbrace{\exp\ldots\exp}_{j} m_\alpha, \]

if

\[ m=\lim_{\alpha\in A} m_\alpha \]

and \(m_\alpha<m\). Further, put

\[ \psi_j(m)=\underbrace{\exp\ldots\exp}_{j} m \quad\text{and}\quad \lg_j(m)=\underbrace{\log\ldots\log}_{j} m. \]

Lemma 1. Let \(m_1\) be an infinite cardinal number. Put \(\varphi_2(m_1)=m_2\), and let \(m_3\) be the least cardinal number greater than \(m_2\). Let

\[ F_1 \supset F_2 \supset \ldots \supset F_\alpha \supset \ldots,\quad \alpha<\omega(m_3), \]

be a strictly decreasing well-ordered sequence of closed sets of a Hausdorff space \(X\). Then \(X\) contains a discrete subspace \(T\) of cardinality \(m_1\).

Proof. Put \(\Phi_\alpha=F_\alpha\setminus F_{\alpha+1}\). We note that all \(\Phi_\alpha\), \(1\le \alpha\le \omega(m_3)\), are nonempty and pairwise disjoint sets. The construction of a discrete subspace \(T\subset X\) of cardinality \(m_1\) will be carried out by transfinite induction.

Take some point \(x_1\in\Phi_1\) and put \(Ox_1=X\setminus F_2\). Suppose that, for some ordinal \(\xi<\omega(m_1)\), points

\[ x_1,x_2,\ldots,x_\beta,\ldots,\quad \beta<\xi, \]

have been constructed such that \(x_\beta\in\Phi_{\alpha(\beta)}\) and

\[ \alpha(1)<\alpha(2)<\ldots<\alpha(\beta)<\ldots, \]

and, moreover, for every point \(x_\beta\), \(\beta<\xi\), a neighborhood \(Ox_\beta\) has been found possessing the following two properties: a) \(Ox_\beta\not\ni x_\gamma\), if \(\gamma<\beta\); b) \(Ox_\beta\cap F_{\alpha(\beta)+1}=\varnothing\). Consider the space \(\Phi=[R]_X\), if \(R=\)

* We note that the basic identity of logarithms in the domain of real numbers turns into an inequality in the domain of cardinal numbers, i.e. \(\exp\log \ge\). For example, if \(n=\aleph_0\), then \(\log \aleph_0=\aleph_0\) and \(\exp \aleph_0>\aleph_0\).

** One says that a space \(X\) satisfies Suslin’s condition \((^1)\) if every system of disjoint (i.e. pairwise nonintersecting) open sets in \(X\) is at most countable.

\[ = \bigcup_{\beta<\zeta} x_\beta . \]
We note that \(R\) is everywhere dense in \(\Phi\), and
\[ |R|=\left|\bigcup_{\beta<\zeta} x_\beta\right|=|\zeta|<\mathfrak m_1{}^{*}. \]
Hence, by Pospíšil’s theorem \((^{4})\), we obtain
\[ |\Phi|\le \exp \exp |\zeta|\le \varphi_2(\mathfrak m_1)=\mathfrak m_2. \]
Since all the \(\Phi_\alpha\) are disjoint, while the cardinality of the family \(\{\Phi_\alpha\}\), \(\alpha<\omega(\mathfrak m_3)\), is equal to \(\mathfrak m_3\), is regular, and \(\mathfrak m_3>\mathfrak m_2\), there exists a least ordinal number \(\theta<\omega(\mathfrak m_3)\) such that
\[ \Phi_\alpha\cap \Phi=\varnothing \]
for all \(\alpha\ge \theta\). In this case put \(\theta=\alpha(\zeta)\) and choose some point
\[ x_\zeta\in \Phi_{\alpha(\zeta)}. \]
Let
\[ Ox_\zeta=(X\setminus \Phi)\cap (X\setminus F_{\alpha(\zeta)+1}). \]
We note that, if \(\beta<\zeta\), then all points \(x_\beta\) lie in \(\Phi\), and therefore \(Ox_\zeta\not\ni x_\beta\), \(\beta<\zeta\), i.e. a) is satisfied. On the other hand, by definition,
\[ Ox_\zeta\cap F_{\alpha(\zeta)+1}=\varnothing, \]
i.e. b) is satisfied. Moreover, for every \(\beta<\zeta\) we have
\[ \alpha(\beta)<\alpha(\zeta), \]
since \(\Phi_\alpha\cap \Phi=\varnothing\) for all \(\alpha\ge \theta=\alpha(\zeta)\). Thus, for every \(\beta<\omega(\mathfrak m_1)\) we can construct points \(x_\beta\) so that conditions a) and b) are fulfilled. The induction is complete.

Let
\[ T=\{x_\beta\},\qquad 1\le \beta<\omega(\mathfrak m_1). \]
Since \(x_\beta\in \Phi_{\alpha(\beta)}\) and all the \(\Phi_\alpha\) are disjoint, we have \(|T|=\mathfrak m_1\). We show that \(T\) is a discrete subspace of \(X\). Indeed, for an arbitrary point \(x_\beta\) consider the neighborhood \(Ox_\beta\) constructed above. By a), this neighborhood contains no point \(x_\gamma\) if \(\gamma<\beta\). If, however, \(\gamma>\beta\), then \(\alpha(\gamma)>\alpha(\beta)\), and therefore \(\alpha(\gamma)\ge \alpha(\beta)+1\); hence
\[ x_\gamma\in \Phi_{\alpha(\gamma)}\subset F_{\alpha(\gamma)}\subset F_{\alpha(\beta)+1}. \]
Therefore, by b), \(Ox_\beta\not\ni x_\gamma\). Thus
\[ T\cap Ox_\beta=x_\beta. \]
This means that \(T\) is discrete. The lemma is proved.

As is known, a topological space is called scattered if it contains no dense-in-itself (i.e. perfect) subset.

Lemma 2\({}^{**}\). Let \(\mathfrak m\) be an infinite cardinal number and let
\[ F_1\supset F_2\supset \cdots \supset F_\alpha\supset \cdots,\qquad \alpha<\omega(\mathfrak m), \]
be a strictly decreasing well-ordered sequence of closed sets of a Hausdorff space \(X\). Then \(X\) contains a scattered subspace \(R\) of cardinality \(\mathfrak m\).

Proof. We note that, for each \(\alpha\),
\[ F_\alpha\setminus F_{\alpha+1}=\Phi_\alpha \]
are nonempty pairwise disjoint sets. Choose one point
\[ x_\alpha\in \Phi_\alpha. \]
Denote the resulting set by \(R\). It is asserted that the set \(R\) is a scattered subspace of \(X\). Indeed, suppose that \(R\) contains a dense-in-itself subset \(M\subset R\). Let
\[ x_\nu\in M \]
be the point with the least index among those belonging to \(M\). Then, on the one hand,
\[ x_\nu\in [M\setminus x_\nu]_X, \]
and on the other hand
\[ x_\nu\in \Phi_\nu=F_\nu\setminus F_{\nu+1}. \]
Since \(\nu\) is the least index, \(M\setminus x_\nu\subset F_{\nu+1}\), i.e.
\[ [M\setminus x_\nu]_X\subset F_{\nu+1}, \]
which is impossible, since \(x_\nu\notin F_{\nu+1}\). The lemma is proved.

Lemma 3. Let \(\mathfrak m\) be an infinite cardinal number. Suppose that every strictly decreasing well-ordered sequence
\[ F_1\supset F_2\supset \cdots \supset F_\alpha\supset \cdots,\qquad \alpha<\omega(\mathfrak m), \]
of closed sets of a Hausdorff space \(X\) is stationary beginning with some \(\alpha_0<\omega(\mathfrak m)\). Then
\[ |X|\le \varphi_1(\mathfrak m). \]

This lemma, for the case \(\mathfrak m=\aleph_1\), was first proved by P. S. Alexandrov and P. S. Urysohn \((^{6})\). For the case \(\mathfrak m=\aleph_{\rho+1}\), it was proved by Yu. M. Smirnov \((^{7})\). In the case where
\[ \mathfrak m=\lim_{\alpha}\mathfrak m_\alpha \]
and \(\mathfrak m_\alpha<\mathfrak m\), we note that, since for an arbitrary cardinal number \(\mathfrak m_\alpha<\mathfrak m\) the cardinality of all possible sequences of 0’s and 1’s of length \(\mathfrak m_\alpha\) is equal to \(\exp \mathfrak m_\alpha\), the cardinality of all sequences of 0’s and 1’s of arbitrary length less than \(\mathfrak m\) does not exceed
\[ \sum_\alpha \exp \mathfrak m_\alpha=\varphi_1(\mathfrak m). \]
Further, since each point \(x\in X\) is completely determined by some such sequence (see the proof of Theorem 6 in \((^{7})\), p. 177), it follows that
\[ |X|\le \varphi_1(\mathfrak m). \]

As is known, a cardinal number \(\mathfrak m>1\) is called a caliber (N. A. Shanin) of a space \(X\) if every family of cardinality \(\mathfrak m\), consisting of

\[ {}^{*}\ |\xi|\ \text{denotes the cardinality of all ordinal numbers less than } \xi. \]

\[ {}^{**}\ \text{This lemma for Fréchet–Urysohn spaces was first proved by V. Sierpiński }(^{5}). \]

consisting of nonempty open subsets of \(X\), has an equipotent subfamily with nonempty intersection. We shall call a topological space \(X\) separated if \(|X|\) is not a caliber of \(X\). Let us note that every discrete space is scattered and separated. However, there exist scattered spaces that are not separated, and, conversely, there exist separated spaces that are not scattered. We record without proof the following fact. There exists a separated Hausdorff space \(X\) of cardinality \(\aleph_1\) satisfying Suslin’s condition.

Let \(X\) be a Hausdorff space. Denote by \(\mathfrak B_i\), respectively, the set of all discrete (\(i=1\)), separated (\(i=2\)), or scattered (\(i=3\)) subspaces of \(X\). To each \(Y_\alpha \in \mathfrak B_i\) assign the cardinal number \(|Y_\alpha|\). Since the cardinality of a space is a weight characteristic in the sense of paper \((^8)\), this correspondence, just defined, is a weight function \(l_i(Y_\alpha)\) defined on the set \(\mathfrak B_i\). Denote by \(\mathfrak a_i=\sup l_i(Y_\alpha)=\sup |Y_\alpha|\), where \(Y_\alpha \in \mathfrak B_i\).

Theorem. For every Hausdorff space \(X\) we have \(|X|\leqslant \psi_{4-i}(\mathfrak a_i)\), \(i=1,2,3\).

Proof. We consider successively three cases.

First case (\(i=1\)). Put \(\mathfrak a_1=\aleph_0\), \(\mathfrak m_1=\aleph_{\rho+1}\), \(\mathfrak m_2=\varphi_2(\mathfrak m_1)\), and let \(\mathfrak m_3\) be the least cardinal number greater than \(\mathfrak m_2\). Note that every strictly decreasing well-ordered sequence
\[ F_1 \supset F_2 \supset \cdots \supset F_\alpha \supset \cdots,\quad \alpha<\omega(\mathfrak m_3), \]
consisting of closed subsets of \(X\), is stationary beginning with some \(\alpha_0<\omega(\mathfrak m_3)\), for otherwise, by Lemma 1, \(X\) would contain a discrete subspace \(T\) of cardinality \(\mathfrak m_1\), which is impossible, since \(\mathfrak m_1>\mathfrak a_1\). Hence, by Lemma 3, we have
\[ |X|\leqslant \varphi_1(\mathfrak m_3)=\exp \mathfrak m_2=\exp \varphi_2(\mathfrak m_1)=\exp \exp \exp \mathfrak a_1=\psi_3(\mathfrak a_1). \]

Second case (\(i=2\)). In this case we shall prove that the density \(sX^*\) does not exceed \(\mathfrak a_2\), whence, by Pospíšil’s theorem \((^4)\), it will follow that \(|X|\leqslant \exp \exp \mathfrak a_2=\psi_2(\mathfrak a_2)\). Suppose that \(sX>\mathfrak a_2\); then \(sX\geqslant \mathfrak n\), where \(\mathfrak n\) is the least cardinal number greater than \(\mathfrak a_2\). We construct a separated space \(Y\subset X\), \(|Y|=\mathfrak n\). Let
\[ X=\{x_1,x_2,\ldots,x_\mu,\ldots,\ \mu<\omega(|X|)\} \]
be well ordered. Put \(y_1=x_1\). Suppose that, for some \(\alpha<\omega(\mathfrak n)\), points \(y_1,y_2,\ldots,y_\beta,\ldots\), \(\beta<\alpha\), belonging to \(Y\), have been constructed in such a way that, for every \(\beta<\alpha\), we have
\[ y_\beta\notin\Bigl[\bigcup_{\gamma<\beta} y_\gamma\Bigr]_X. \]
Consider
\[ F=\Bigl[\bigcup_{\beta<\alpha} y_\beta\Bigr]_X. \]
Since \(|\alpha|\leqslant \mathfrak a_2\), while \(sX\geqslant \mathfrak n>\mathfrak a_2\), we have \(X\setminus F\ne\varnothing\). Choose the least index \(\mu<\omega(|X|)\) such that \(x_\mu\in X\setminus F\), and put \(x_\mu=y_\alpha\). The induction is complete. Let \(Y\) be the set of all chosen \(y\)’s. Then, by construction, \(|Y|=\mathfrak n\). Denote
\[ U_\alpha=Y\setminus\Bigl[\bigcup_{\beta<\alpha} y_\beta\Bigr]_X. \]
Note that all \(U_\alpha\ne\varnothing\) and that
\[ U_1 \supset U_2 \supset \cdots \supset U_\alpha \supset \cdots,\quad \alpha<\omega(\mathfrak n), \]
is a strictly decreasing well-ordered sequence of open sets with empty intersection, i.e. \(|Y|\) is not a caliber of \(Y\). This means that \(Y\) is a separated space of cardinality \(\mathfrak n\), which is impossible, since \(\mathfrak n>\mathfrak a_2\). Hence \(sX\leqslant \mathfrak a_2\), and, as was said above, this means that \(|X|\leqslant \psi_2(\mathfrak a_2)\).

Third case (\(i=3\)). Let \(\mathfrak m\) be the least cardinal number greater than \(\mathfrak a_3\). Then every strictly decreasing well-ordered sequence of closed subsets of \(X\) is stationary beginning with some \(\alpha_0<\omega(\mathfrak m)\), for otherwise, by Lemma 2, there would exist in \(X\) a scattered subspace of cardinality \(\mathfrak m\), which is impossible. Applying Lemma 3, we obtain
\[ |X|\leqslant \varphi_2(\mathfrak m)=\exp \mathfrak a_3. \]
The theorem is completely proved.

* The density \(sX\) of a space \(X\) is the minimum of the cardinalities of everywhere dense subsets of \(X\).

This theorem had already been proved when the very recent note of Isbell \((^9)^*\) appeared, in which, for a completely regular \(X\) and \(i = 1\), an analogous result was obtained. We note that Isbell’s method is not suitable for Hausdorff spaces.

Corollary 1. For a Hausdorff space \(X\) we have the inequality
\[ \mathfrak a_i=\sup_{Y_\alpha\in \mathfrak B_i} l_i(Y_\alpha)\geq \lg_{4-i}|X|,\qquad i=1,2,3. \]

Indeed, suppose, for example, that \(\mathfrak a_3<\log |X|\); then, by the definition of the logarithm, \(\exp \mathfrak a_3<|X|\), which contradicts the theorem \((i=3)\). The cases \(i=2\) and \(i=1\) are considered analogously.

Corollary 2. If a Hausdorff space \(X\) satisfies Suslin’s condition hereditarily, then
\[ |X|\leq \psi_3(\aleph_0). \]

I do not know whether this estimate for the cardinality of \(X\) is sharp.

In conclusion, the author expresses gratitude to A. Mishchenko for discussing this work and for a number of valuable remarks.

Moscow State University
named after M. V. Lomonosov

Received
17 II 1965

REFERENCES

\(^1\) M. Suslin, Fundam. Math., 1, 223 (1920).
\(^2\) E. Marczewski, Fundam. Math., 34, 127 (1947).
\(^3\) B. Efimov, DAN, 149, 1011 (1963).
\(^4\) B. Pospíšil, Čas. pro pěst. mat. fys., 67, 89 (1938).
\(^5\) W. Sierpiński, Fundam. Math., 2, 181 (1921).
\(^6\) P. S. Alexandrov, P. S. Urysohn, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 31 (1950).
\(^7\) Yu. M. Smirnov, Izv. AN SSSR, 14, No. 2, 155 (1950).
\(^8\) B. Efimov, DAN, 158, 1260 (1964).
\(^9\) I. R. Isbell, Czechoslovak Math. J., 14 (89) (1964).
\(^10\) I. I. Parovichенко, DAN, 115, 1074 (1957).

\(^*\) The second part of the assertion of Theorem 1 in note \((^9)\) is erroneous, since there exist \((^{10})\), for every number \(m\geq\aleph\), completely regular spaces of cardinality \(m\) and weight \(\exp m\), which therefore cannot be embedded in \(I^m\).