UDC 519.50+519.54

MATHEMATICS

S. NEDEV

SYMMETRIZABLE SPACES

AND FINAL COMPACTNESS

(Presented by Academician P. S. Aleksandrov, 4 X 1966)

All the proofs in this article are based on Theorem 1; its formulation is based on the notion, recently proposed by A. V. Arhangel’skii, of a weakly discrete system of sets. The proof of Theorem 1 rests on a technique developed by Bing for the case of spaces with a refining sequence of covers and subsequently applied by MacAuley in the study of strongly symmetrizable spaces.

The proof of Theorem 2 is based on the main idea of the proof of the theorem on the metrizability of a symmetrizable bicompactum, due to A. V. Arhangel’skii and V. Stoyanovsky (see (¹)). Theorem 2 may be regarded as a partial answer to Michael’s question: is every closed set in a symmetrizable space a set of type \(G_\delta\).

Definition 1 (A. Arhangel’skii). A system \(\{M_\alpha\}_{\alpha\in A}\) of subsets of a topological space \(X\) is called weakly discrete if from \(x_\alpha \in M_\alpha\) it follows that the system of points \(\{x_\alpha\}_{\alpha\in A}\) is discrete in \(X\).

Theorem 1. In every open cover of a symmetrizable space one can inscribe a \(\sigma\)-weakly discrete cover.

Proof. Let \(X\) be a space with symmetrix \(d\). We shall call a system of subsets \(\{M_\alpha\}_{\alpha\in A}\) discrete with respect to the symmetrix \(d\) if for every \(x\in X\) there exists \(\varepsilon=\varepsilon(x)>0\) such that \(O_\varepsilon x\) meets no more than one element of the system \(\{M_\alpha\}_{\alpha\in A}\).

Lemma. If the system \(\{M_\alpha\}_{\alpha\in A}\) is discrete with respect to the symmetrix \(d\) and, for every \(\alpha\in A\), \(M_\alpha\) is closed in \(X\), then the system \(\{M_\alpha\}_{\alpha\in A}\) is discrete in \(X\).

Indeed, if for some \(A'\subset A\) the set \(\bigcup_{\alpha\in A'} M_\alpha\) is not closed in \(X\), then there exists a point
\[ x\in X\setminus \bigcup_{\alpha\in A'} M_\alpha \]
such that
\[ d\!\left(x,\bigcup_{\alpha\in A'} M_\alpha\right)=0. \]
But the system \(\{M_\alpha\}_{\alpha\in A}\) is discrete with respect to the symmetrix \(d\); consequently,
\[ d(x,M_{\alpha_0})=0 \]
for some \(\alpha_0\in A'\). Hence \(x\in M_{\alpha_0}\), and this contradicts the condition
\[ x\in X\setminus \bigcup_{\alpha\in A'} M_\alpha . \]

Let now \(\omega=\{U_\alpha\}_{\alpha\in A}\) be an open cover of the space \(X\), with \(A\) a well-ordered set. For each natural \(n\) and each \(\alpha\in A\), put
\[ M_\alpha^n=\mathscr{E}\{x\in X,\ x\in U_\alpha \setminus \bigcup_{\beta<\alpha} U_\beta,\ O_{1/n}x\subset U_\alpha\}. \]

We shall show that, for each fixed \(n\), the system \(\{M_\alpha^n\}_{\alpha\in A}\) is discrete with respect to the symmetrix \(d\). Indeed, let \(x\in X\). Then, for some \(\alpha\in A\),
\[ x\in U_\alpha\setminus \bigcup_{\beta<\alpha} U_\beta \]
and, for some \(m\ge n\),
\[ O_{1/m}x\subset U_\alpha \]
(whence it is clear that \(\{M_\alpha^n\}_{\alpha\in A,\ n=1,2,\ldots}\) is a cover). Suppose that for some \(\beta\),
\[ O_{1/m}x\cap M_\beta^n\ne \Lambda . \]
This is impossible for \(\beta>\alpha\), since in that case
\[ M_\beta^n\subset U_\beta\setminus U_\alpha, \]
whereas
\[ O_{1/m}x\subset U_\alpha . \]
But \(O_{1/m}x\cap M_\beta^n\ne \Lambda\) also cannot occur for \(\beta<\alpha\), since then, from
\[ y\in O_{1/m}x\cap M_\beta^n, \]
it follows that
\[ x\in O_{1/n}y\subset O_{1/m}y\subset U_\beta, \]
which contradicts the relation
\[ x\in U_\alpha\setminus \bigcup_{\gamma<\alpha} U_\gamma \subset U_\alpha\setminus U_\beta . \]

To complete the proof of Theorem 1 it remains only to apply the lemma.

Theorem 2. A finally compact symmetrizable space is hereditarily finally compact.

Proof. Let \(X\) be a Hausdorff finally compact space with a symmetric \(d\), and let \(F\) be a closed subspace of the space \(X\). Then \(X' = X \setminus F\) is a symmetrizable space, as an open subspace of a symmetrizable space. Let \(\{U_\alpha\}_{\alpha \in A}\) be an open cover of the space, where \(A\) is a well-ordered set. Put again
\[ M_\alpha^n=\mathcal E\{x\in X',\ x\in \overline{U}_\alpha\setminus \bigcup_{\beta<\alpha}U_\beta,\ O_{1/n}x\subset U_\alpha\} \]
and suppose that for some \(n\) the set of nonempty elements of the system \(\{M_\alpha^n\}_{\alpha\in A}\) is uncountable. Choose from each nonempty element of the system \(\{M_\alpha^n\}_{\alpha\in A}\) a point \(x_\alpha \in M_\alpha^n\).

Since
\[ F=\bigcap_{m=1}^{\infty} O_{1/m}F, \]
there exist \(\varepsilon>0\) and \(A'\subset A\) such that:

1) \(x_\alpha\notin O_\varepsilon F\) for \(\alpha\in A'\), and 2) the system \(\{x_\alpha\}_{\alpha\in A'}\) is uncountable.

The set \(\mathcal E\{x_\alpha:\ \alpha\in A'\}\) is not closed in \(X\), because, as we saw in the proof of Theorem 1, it is discrete in the space \(X'\), and it has limit points by virtue of the final compactness of the space \(X\) and the large cardinality of the set \(\mathcal E\{x_\alpha:\ \alpha\in A'\}\). Consequently, there exists a point \(x_0\in X\setminus \mathcal E\{x_\alpha:\ \alpha\in A'\}\) such that \(d(x_0,\mathcal E\{x_\alpha:\ \alpha\in A'\})=0\).

From the last equality and the definition of the set \(A'\) it follows that \(x_0\in X'\), but this contradicts the discreteness of the system \(\{x_\alpha\}_{\alpha\in A'}\) in \(X\). The contradiction completes the proof of Theorem 2, since it shows that from the cover \(\{M_\alpha^n\}_{\alpha\in A,\ n=1,2,\ldots}\), and consequently also from the cover \(\{U_\alpha\}_{\alpha\in A}\), one can choose a countable subcover.

Theorem 3. A symmetrizable space is finally compact if and only if every discrete system of points in it is countable.

Proof follows directly from Theorem 1, the lemma, and the known fact that in any finally compact space every discrete system is countable.

Corollary 1. A Hausdorff symmetrizable hereditarily separable space is hereditarily finally compact \(*\).

Corollary 2. A Hausdorff countably compact symmetrizable space is compact \(**\).

Proof. If \(X\) is a countably compact space, then in \(X\) every discrete system of points is finite. Hence, if \(X\) is a symmetrizable space, then by Theorem 3 it follows that \(X\) is finally compact and, consequently, bicompact. By the main theorem of Arhangel’skii and Stoyanovskij formulated at the beginning \((^1)\), \(***\) a symmetrizable bicompactum is compact.

Theorem 4. A symmetrizable \(M\)-space (in the sense of Morita \((^6)\)) is metrizable.

Proof. By Morita’s theorem \(((^6),\) Theorem 5.3), every \(M\)-space admits a closed mapping onto a metric space under which the preimage of each point is a countably compact space. Hence, from Corollary 2 we obtain that every symmetrizable \(M\)-space is a feathered paracompactum (see \((^3)\)), and an arbitrary feathered paracompactum with a symmetric is metrizable, by a theorem of A. V. Arhangel’skii.

\(*\) If the word symmetrizable is replaced by strongly symmetrizable, one obtains a statement proved by Sider \((^7)\). His arguments do not carry over to the case of symmetrizable spaces.

\(**\) If the word symmetrizable is replaced by strongly symmetrizable, one obtains a statement proved by Nemytskii \((^5)\). His arguments do not carry over to the case of symmetrizable spaces.

\(***)\) The key role in the proof of this theorem is played by the special case of Theorem 2 concerning bicompacta. Then a known theorem of Nemytskii is applied \((^5)\).

Definition 2 (A. Arhangel’skii (2)). A symmetric \(d\) of a symmetrizable space \(X\) is called a symmetric satisfying the weak Cauchy condition* if, from \(A \subset X\) and \([A]\ne A\), it follows that for every \(\varepsilon>0\) there are distinct points \(x\) and \(y\) in \(A\) for which \(d(x,y)<\varepsilon\).

In Arhangel’skii’s paper (2) the following assertion is proved: a factor, weakly uniformly** image of a metric space is symmetrizable with a symmetric satisfying the weak Cauchy condition.

Using the ideas of (2), we prove the following theorem.

Theorem 5. If the Hausdorff symmetric \(d\) of a finally compact symmetrizable space \(X\) satisfies the weak Cauchy condition, then for every subspace \(X'\) of the space \(X\) and every \(\varepsilon>0\), from the cover \(\{O_{\varepsilon}x\}_{x\in X'}\) of the space \(X'\) one can choose a countable cover of the space \(X'\).

Proof. Starting with an arbitrary point \(x_0\in X'\), we construct a system of points \(\{x_\alpha\in X'\}_{\alpha\in A}\) such that: 1) \(A\) is well ordered; 2)
\[ X'=\bigcup_{\alpha\in A} O_\varepsilon x_\alpha; \]
3)
\[ x_\alpha\in X'\setminus \bigcup_{\beta<\alpha} O_\varepsilon x_\beta . \]
The last condition can obviously be replaced by 3′) for \(\alpha\ne\beta\), \(d(x_\alpha,x_\beta)\ge \varepsilon\). The system of points \(\{x_\alpha\}_{\alpha\in A}\) so constructed turns out to be discrete in \(X\). Indeed, if we suppose that for some subset \(A'\) of the set \(A\) the set \(\mathscr E\{x_\alpha:\alpha\in A'\}\) is not closed in \(X\), then we would obtain that for some \(\alpha\ne\beta\), \(d(x_\alpha,x_\beta)<\varepsilon\), which contradicts condition 3′). Consequently, the system of points \(\{x_\alpha\}_{\alpha\in A}\) is countable, as is every discrete system of points in a finally compact space. The theorem is proved.

Corollary 3. A symmetrizable finally compact space, some symmetric of which satisfies the weak Cauchy condition, is hereditarily separable.

Proof. Let a symmetric \(d\) of the finally compact symmetrizable space \(X\) satisfy the weak Cauchy condition, and let \(X'\) be an arbitrary subspace of the space \(X\). By Theorem 5, for every natural \(n\) there exists a sequence of points \(\{x_m^n\}_{m=1}^{\infty}\) such that: 1) \(x_m^n\in X'\) for every \(m\), and 2)
\[ X'=\bigcup_{m=1}^{\infty} O_{1/n}x_m^n . \]
It is easy to verify that the countable set \(\mathscr E\{x_m^n,\ n,m=1,2,\ldots\}\) is everywhere dense in \(X'\).

The author expresses his deep gratitude to A. V. Arhangel’skii for his help and for providing the opportunity to become acquainted with his unpublished works.

Moscow State University
named after M. V. Lomonosov

Received
28 VI 1966

REFERENCES

1. A. V. Arhangel’skii, UMN, 21, no. 4 (130), 133 (1966).
2. A. V. Arhangel’skii, DAN, 164, no. 2 (1965).
3. A. V. Arhangel’skii, Matem. sborn., 67 (109), no. 1 (1965).
4. P. S. Aleksandrov, V. V. Nemytskii, Matem. sborn., 3, 3, 665 (1938).
5. V. Niemytzki, Math. Ann., 101, no. 5, 666 (1931).
6. K. Morita, Proc. Japan Acad., 39, 150 (1963).
7. J. G. Geder, Pacific J. Math., 11, no. 1 (1961).

* A symmetric \(d\) is called a Cauchy symmetric (see (4)) if, from
\[ \lim_{n\to\infty} d(x_n,x)=0, \]
it follows that
\[ \lim_{n\to\infty} d(x_n,x_{n+1})=0. \]

* A mapping \(f:X\to Y\) of a symmetrizable space \(X\) onto a topological space \(Y\) is called weakly uniformly continuous* if for any point \(y\in Y\) and its neighborhood \(U\),
\[ \rho(f^{-1}y,\ X\setminus f^{-1}U)>0, \]
where \(\rho\) is a fixed symmetric of the space \(X\).