MATHEMATICS

S. ILIADIS

CHARACTERIZATION OF SPACES BY MEANS OF \(H\)-CLOSED EXTENSIONS*

(Presented by Academician P. S. Aleksandrov on 29 X 1962)

In the paper \((^1)\) Katětov posed several questions concerning the maximal \(H\)-closed extension \(\tau R\).

Here are two of them:

1. Under what conditions imposed on the space \(R\) do the extension \(\tau R\) and the Čech extension \(\beta R\) coincide? The answer to this question is given by Theorem 12 of \((^3)\).

2. It is known that if completely regular spaces \(R_1\) and \(R_2\) satisfy the first axiom of countability, then from a homeomorphism of the extensions \(\beta R_1\) and \(\beta R_2\) there follows a homeomorphism of the spaces \(R_1\) and \(R_2\).

It would be interesting to find sufficiently broad conditions for analogous assertions concerning the extension \(\tau R\).

Here a new proof is given of Theorem 12 of \((^3)\). In Theorem 5 there are given, as it seems to me, the conditions required in the second question. In conclusion, some properties are considered of points of a special type, called here “maximal.”

§ 1. A point \(x\) of the space \(R\) will be called a point of contact of a set \(M\), lying in \(R\), if \(x \notin M\), \(x \in \overline{M}\) and \(x \in R \setminus \overline{M}\). If some non-isolated point \(\xi\) is not a point of contact for any set, then we shall call it a maximal point**.

Theorem 1. If \(\xi \in \tau R \setminus R\), then \(\xi\) is maximal in \(\tau R\).

Proof. First let us recall how the space \(\tau R\) is constructed (see \((^1)\)). A system \(\gamma\), consisting of nonempty open sets \(\Gamma \subset R\), will be called an \(\alpha\)-system if it satisfies the following two conditions: 1) from the inclusions \(\Gamma_1 \in \gamma\) and \(\Gamma_2 \in \gamma\) there follows the inclusion \(\Gamma_1 \cap \Gamma_2 \supset \gamma\); 2)

\[ \bigcap_{\Gamma \in \gamma} \overline{\Gamma} = \varnothing^{***}. \]

As usual, an \(\alpha\)-system will be called maximal if every \(\alpha\)-system containing it coincides with it.

The points of the space \(\tau R\) are all points of the space \(R\) and all maximal \(\alpha\)-systems \(\xi\). A base of the space \(\tau R\) at a point \(x \in R\) is formed by the system of neighborhoods of the point \(x\) in \(R\). A base of the space \(\tau R\) at a point \(\xi\) is formed by the sets \(\xi \cup \Gamma\), where \(\Gamma \in \xi\). It is not hard to show (see \((^1)\)) that \(\tau R\) is the maximal \(H\)-closed extension of the space \(R\). From the construction it is seen that \(R\) is open in \(\tau R\), and \(\tau R \setminus R\) is discrete.

We now prove the theorem. Let \(\xi \in \tau R \setminus R\). Suppose that there exists a set \(M\) such that \(\xi \notin M\), \(\xi \in \overline{M}\), and \(\xi \in \tau R \setminus \overline{M}\) (here \(\overline{M}\) is the closure in \(\tau R\)). The set \(\tau R \setminus \overline{M}\) is open in \(\tau R\). From the maximality of the system \(\xi\) it follows that

\[ R \cap (\tau R \setminus \overline{M}) \in \xi, \]

since \(R \cap (\tau R \setminus \overline{M})\) intersects every neighborhood of the point \(\xi\). But this contradicts the condition \(\xi \in \overline{M}\). The theorem is proved.

* All spaces considered are Hausdorff. An extension \(tR\) of a space \(R\) is called \(H\)-closed if it is closed in every (topological) space containing it.

** \(\overline{M}\) is the closure of the set \(M\) in \(R\).

*** \(\varnothing\) is the empty set.

Definition 1 (see (2)). An infinite set \(E\) of the space \(R\) converges in cardinality to a point \(x\) of the space \(R\), if for every neighborhood \(Ox\) we have the inequality
\[ \operatorname{card}(E\cap Ox)>\operatorname{card}(E\setminus Ox). \]

Theorem 2. No set \(E\) of the space \(R\) of regular cardinality can converge in cardinality to any maximal point \(\xi\) of the space \(R\).

Proof. Let the set \(E\) be of regular cardinality \(\aleph_\tau\) and converge in cardinality to the maximal point \(\xi\). Write all points of the set \(E\) in the form of a sequence of type \(\omega_\tau\):
\[ x_1,x_2,\ldots,x_\lambda,\ldots;\qquad \lambda<\omega_\tau . \]
By \(\Phi_\lambda\) denote the closure of the set of all points \(x_{\lambda'}\) for which \(\lambda'\geq \lambda\). We first prove that \(\xi\in\langle\Phi_\lambda\rangle\)* for every \(\lambda\).

Indeed, if \(\xi\notin\langle\Phi_\lambda\rangle\), then \(\xi\in R\setminus\Phi_\lambda\). Let, further, \(M=\Phi_\lambda\setminus\xi\). It is clear that \(\xi\notin M\), \(\xi\in\overline M=\Phi_\lambda\). All this contradicts the fact that \(\xi\) is a maximal point. Hence \(\xi\in\langle\Phi_\lambda\rangle\).

To complete the proof we shall need the following

Lemma 1. For every index \(\lambda\) there is an index \(\alpha\) such that
\[ \langle\Phi_\lambda\rangle\setminus\Phi_\alpha\ne\varnothing . \]

Proof. Let \(x\ne\xi\) and \(x\in\langle\Phi_\lambda\rangle\). Take disjoint neighborhoods \(O\xi\) and \(Ox\) of the points \(\xi\) and \(x\).

Recall that \(\operatorname{card}(E\setminus O\xi)<\operatorname{card}E\); consequently, by regularity of the number \(\omega_\tau\), there is an \(\alpha\) such that \(\lambda'<\alpha\) as soon as \(x_{\lambda'}\in R\setminus O\xi\). Therefore
\[ \Phi_\alpha\subseteq\overline{O\xi}\subseteq R\setminus Ox, \]
and hence \(x\in\langle\Phi_\lambda\rangle\setminus\Phi_\alpha\). The lemma is proved.

We continue the proof of the theorem. Take a sequence
\[ \Phi_{\lambda_1},\Phi_{\lambda_2},\ldots,\Phi_{\lambda_\alpha},\ldots, \]
such that
\[ \langle\Phi_{\lambda_\alpha}\rangle\setminus\Phi_{\lambda_{\alpha+1}}\ne\varnothing \]
for every \(\alpha\), and such that the sequence of all indices \(\lambda_\alpha\) is cofinal with the sequence
\[ 0,1,2,\ldots,\lambda,\ldots,\qquad \lambda<\omega_\tau . \]
Such a sequence exists, as follows from the lemma. Let
\[ H_\alpha=\langle\Phi_{\lambda_\alpha}\rangle\setminus\Phi_{\lambda_{\alpha+1}}. \]
If \(\alpha'\ne\alpha''\), then \(H_{\alpha'}\cap H_{\alpha''}=\varnothing\). In the system \(\eta=\{H_\alpha\}\), consisting of all sets \(H_\alpha\), consider the subsystem \(\eta_0\) consisting of all those sets \(H_\alpha\) for which \(\alpha=\alpha'+n\), where \(n\) is an even natural number, and \(\alpha'\) is a limit transfinite number or \(0\). Let
\[ U=\bigcup_{H_\alpha\in\eta_0}H_\alpha \qquad\text{and}\qquad V=\bigcup_{H_\alpha\notin\eta_0}H_\alpha . \]
The sets \(U\) and \(V\) are nonempty, open, and disjoint. It is clear that \(\xi\notin U\) and \(\xi\notin V\). We shall show that
\[ O\xi\cap\bigl(\langle\Phi_{\lambda_\alpha}\rangle\setminus\Phi_{\lambda_{\alpha+1}}\bigr)\ne\varnothing \]
for every neighborhood \(O\xi\) of the point \(\xi\) and for all indices \(\lambda_\alpha\), starting with some index \(\lambda\) (where \(\lambda\) depends on \(O\xi\)). Let \(O\xi\) be a neighborhood of the point \(\xi\). There exists a \(\lambda\) such that
\[ \Phi_\lambda\subseteq\overline{O\xi}. \]
Let \(\lambda_\alpha>\lambda\). It is clear that
\[ \Phi_{\lambda_\alpha}\subseteq\Phi_\lambda\subseteq\overline{O\xi}. \]
Further, the boundary \(\overline{O\xi}\setminus O\xi\) contains no nonempty open set, while the set
\[ \langle\Phi_{\lambda_\alpha}\rangle\setminus\Phi_{\lambda_{\alpha+1}} \]
is nonempty, open, and is contained in \(O\xi\). But the set
\[ \langle\Phi_{\lambda_\alpha}\rangle\setminus\Phi_{\lambda_{\alpha+1}}=H_\alpha \]
either belongs to \(\eta_0\) or does not belong to \(\eta_0\), depending on the “parity” of the number \(\alpha\). Hence \(\xi\in\overline U\) and \(\xi\in\overline V\). Since \(V\subseteq R\setminus\overline U\), it follows that \(\xi\in R\setminus\overline U\), which is impossible. Theorem 2 is completely proved.

Definition 2 (see (2)). A point \(x\) will be called an \(x\)-point of the space \(R\), if there exists a countable set \(E\) converging to the point \(x\).

Corollary. A maximal point \(\xi\) of the space \(R\) cannot be an \(x\)-point.

We shall now need Theorem 3 and its corollary, due to P. S. Aleksandrov (see (2), Ch. IV, § 1, item 7).

\[ \text{* } \langle M\rangle \text{ is the open kernel of the set } M \text{ in the space } R. \]

Theorem 3. If at a non-isolated point \(x\) the space \(R\) is regular and if \(\psi_x R=\chi_x R^*\), then there exists a set \(E\) of regular cardinality which converges in cardinality to the point \(x\in R\).

Corollary. To every point \(x\) of a bicompactum \(R\) there converges in cardinality some set \(E_x\) of regular cardinality.

From this corollary and from Theorems 1 and 2 it follows immediately:

Theorem 4 (first main theorem). If \(R\) is a completely regular non-bicompact space, then \(\tau R\) and \(\beta R\) do not coincide.

Theorem 5 (second main theorem). Suppose the spaces \(R_1\) and \(R_2\) contain no maximal points. If the extensions \(\tau R_1\) and \(\tau R_2\) are homeomorphic, then the spaces \(R_1\) and \(R_2\) are also homeomorphic.

Proof. Let \(\tau R=\tau R_1=\tau R_2\). The spaces \(R_1\) and \(R_2\) may be regarded as subsets of the space \(\tau R\). We shall need the following

Lemma 2. If the set \(R_1\cup \xi\) is everywhere dense and open in the space \(R\), and the point \(\xi\) is maximal in \(R\), then it is maximal also in \(R_1\cup \xi\).

Proof. Suppose that the point \(\xi\) is maximal in \(R\), but not maximal in \(R_1\cup \xi\). Then there is a subset \(M\) of the set \(R_1\cup \xi\) such that \(\xi\notin M\), \(\xi\in \overline{M}\), and
\[ \xi\in (R_1\cup \xi)\setminus \overline{M}. \]
Here closures are taken in the space \(R_1\cup \xi\). The set
\[ U=(R\cup \xi)\setminus \overline{M} \]
is open in \(R\), and \(\xi\in \overline{U}^{\,R}\). Let \(N=R\setminus (U\cup \xi)\). It is clear that \(\xi\notin N\), \(\xi\in \overline{N}^{\,R}\), but
\[ \xi\in R\setminus \overline{N}^{\,R}, \]
since \(U\subset R\setminus \overline{N}^{\,R}\). This contradicts our assumption. The lemma is proved.

We return to the proof of the theorem. Suppose that \(R_1\setminus R_2\ne \varnothing\), and let \(\xi\in R_1\setminus R_2\). By Theorem 1 the point \(\xi\) is maximal in \(\tau R\), and by Lemma 2 it is maximal in the space \(R_1\cup \xi\), contrary to the assumption. Hence \(R_1\subseteq R_2\). In the same way one proves that \(R_2\subseteq R_1\). Thus \(R_1=R_2\). Theorem 5 is proved.

Corollary. Suppose all points of the spaces \(R_1\) and \(R_2\) are \(\chi\)-points; if the extensions \(\tau R_1\) and \(\tau R_2\) are homeomorphic, then the spaces \(R_1\) and \(R_2\) are also homeomorphic.

§ 2. By \(\psi'_\xi R\) we denote the least of all cardinal numbers \(\psi\) such that the point \(\xi\) has in the space \(R\) a system of neighborhoods \(\{O_\alpha \xi\}\) of cardinality \(\psi\), possessing the property
\[ \bigcap_\alpha \overline{O_\alpha \xi}=\xi . \]

Theorem 6. If \(\xi\) is a maximal point of an \(H\)-closed space \(R\), then
\[ \psi_\xi R\ne \psi'_\xi R . \]

Proof. Let \(\psi_\xi R=\psi'_\xi R\), and let \(\{O_\alpha \xi\}\) be such a system of neighborhoods \(O_\alpha \xi\) of the point \(\xi\), of cardinality \(\psi'_\xi R=\aleph_\tau\), that
\[ \bigcap \overline{O_\alpha \xi}=\xi . \]

Write the elements of this system in a sequence of type \(\omega_\tau\):
\[ O_1\xi,\ O_2\xi,\ldots,\ O_\alpha\xi,\ldots;\qquad \alpha<\omega_\tau . \]
Put \(H_1=O_1\xi\). Suppose that neighborhoods \(H_\beta\) of the point \(\xi\) have been constructed for all \(\beta<\alpha\), possessing the property that, if \(\beta_1\ge \beta_2\), then
\[ \overline{H_{\beta_1}}\subseteq \overline{H_{\beta_2}}. \]
We shall prove that
\[ \bigcap_{\beta<\alpha<\omega_\tau}\overline{H_\beta} \]
contains some neighborhood of the point \(\xi\). The point \(\xi\) cannot be isolated in the intersection \(\bigcap_{\beta<\alpha}\overline{H_\beta}\), for otherwise the cardinality of the system \(\{O_\alpha \xi\}\) would not be equal to \(\psi'_\xi R\).

Let
\[ M=\left(\bigcap_{\beta<\alpha}\overline{H_\beta}\right)\setminus \xi . \]
Since \(\xi\) is not isolated in \(\bigcap_{\beta<\alpha}\overline{H_\beta}\), we have
\[ M=\bigcap_{\beta<\alpha}\overline{H_\beta}. \]
But if \(\bigcap_{\beta<\alpha}\overline{H_\beta}\) contains no neighborhood of the point \(\xi\), then
\[ \xi\in R\setminus \left(\bigcap_{\beta<\alpha}\overline{H_\beta}\right) =R\setminus \overline{M}, \]
which is impossible, since the point \(\xi\) is maximal in \(R\). Hence
\[ \bigcap_{\beta<\alpha}\overline{H_\beta} \]

* \(\chi_xR\) (respectively \(\psi_xR\)) is the least cardinal number which is the cardinality of some base (respectively pseudobase) of the space \(R\) at the point \(x\).

contains a neighborhood of the point \(\xi\). By \(H_\alpha\) we shall denote such a neighborhood of the point \(\xi\), which is contained in \(O_\alpha \xi \cap \left(\bigcap_{\beta<\alpha}\overline{H}_\beta\right)\). It is clear that \(\overline{H}_\alpha \subseteq \overline{H}_\beta\), if \(\alpha \geq \beta\), and that \(\bigcap_\alpha \overline{H}_\alpha=\xi\). For every \(\alpha\) the set \(\langle \overline{H}_\alpha\rangle\) is nonempty. We may assume that \(\langle \overline{H}_\alpha\rangle\setminus \overline{H}_\beta\) is nonempty if \(\beta>\alpha\), for otherwise we could pass to a subsequence having this property. Let \(\xi_\alpha\) be some point belonging to the set \(\langle \overline{H}_\alpha\rangle\setminus \overline{H}_{\alpha+1}\). Since \(\langle \overline{H}_\beta\rangle \supseteq \langle \overline{H}_\alpha\rangle\), if \(\alpha\geq \beta\), the set of points \(\xi_{\alpha'}\) belonging to the set \(R\setminus \langle \overline{H}_\alpha\rangle\) has cardinality less than \(\psi_\xi R=\aleph_\tau\). Let \(x\in R\). Since \(\bigcap_\alpha \overline{H}_\alpha=\xi\), we have \(x\notin \overline{H}_{\alpha_0}\) for some \(\alpha_0\). Hence \(Ox=R\setminus \overline{H}_{\alpha_0}\) is a neighborhood of the point \(x\). But \(\overline{Ox}=R_1\setminus \overline{H}_{\alpha_0}\) is contained in \(R\setminus \langle \overline{H}_{\alpha_0}\rangle\), hence
\[ \operatorname{card}(E\cap \overline{Ox})<\operatorname{card} E, \]
where \(E\) is the set of all points \(\xi_\alpha\). Hence, by virtue of the \(H\)-closedness of the space \(R\), it follows that \(\operatorname{card}(\overline{O\xi}\cap E)=\aleph_\tau\) and \(\operatorname{card}(E\setminus O\xi)<\aleph_\tau\), where \(O\xi\) is any neighborhood of \(\xi\). Let \(G_\alpha=\langle \overline{H}_\alpha\rangle\setminus \overline{H}_{\alpha+1}\). It is clear that \(\xi_\alpha\in G_\alpha\). We divide the system \(\gamma=\{G_\alpha\}\) into two subsystems \(\gamma_1\) and \(\gamma_2\). We put \(G_\alpha\in\gamma_1\) if \(\alpha=\alpha'+n\), where \(n\) is an even natural number, and \(\alpha'\) is a limit transfinite number or \(0\). We set \(U=\bigcup_{G_\alpha\in\gamma_1}G_\alpha\). If \(G_\alpha\notin\gamma_1\), then \(G_\alpha\in\gamma_2\). Let \(V=\bigcup_{G_\alpha\in\gamma_2}G_\alpha\). We shall show that \(O\xi\cap G_\alpha\ne\varnothing\) for any neighborhood \(O\xi\) of the point \(\xi\) and for all indices \(\alpha\), beginning with some index \(\alpha_0\) (\(\alpha_0\) depends on \(O\xi\)). This follows from the fact that \(\overline{O\xi}\) contains all points except, possibly, some set of cardinality less than \(\aleph_\tau\). Hence it follows that \(\xi\in\overline{U}\) and \(\xi\in\overline{V}\). But \(\overline{V}\subseteq R\setminus \overline{U}\), which is impossible by the maximality of the point \(\xi\). The theorem is proved.

Corollary 1. If an \(H\)-closed space \(R\) is regular at its point \(\xi\), then this point is not maximal.

Corollary 2. If \(\xi\) is a maximal point of an \(H\)-closed space \(R\), then \(\psi_\xi R\ne \chi_\xi R\).

It is not difficult to show that \(\psi_x \tau R=\psi_x R\) and \(\chi_x \tau R=\chi_x R\) for all points \(x\in R\).

Corollary 3. If \(\psi_x R_1=\chi_x R_1\) at all points of the space \(R_1\) and \(\psi_x R_2=\chi_x R_2\) at all points of the space \(R_2\), then from the homeomorphism of the extensions \(\tau R_1\) and \(\tau R_2\) there follows a homeomorphism of the spaces \(R_1\) and \(R_2\).

Remark 1. In Corollary 1 one cannot dispense with \(H\)-closedness. Indeed, let the space \(R\) consist of isolated points and let \(\xi\in \tau R\setminus R\). Then the point \(\xi\) in the space \(R\cup \xi\) is maximal, while the space \(R\cup \xi\) is regular at it.

Remark 2. It is not difficult to prove that if the point \(\xi\) is maximal in \(R\), then it is also maximal in \(\tau R\). Since pseudocharacter and character are preserved upon passing to the extension \(\tau R\), Corollary 2 is true not only for \(H\)-closed spaces, but also for any space \(R\).

The author expresses gratitude to Yu. M. Smirnov for supervising the work.

Moscow State University
named after M. V. Lomonosov

Received
20 X 1962

CITED LITERATURE

1. M. Katětov, Čas. matem. fysiky, 72, 17 (1947).
2. P. S. Aleksandrov, P. S. Uryson, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 31 (1950).
3. M. Katětov, Čas. matem. fysiky, 72, 101 (1947).