UDC 513.831

MATHEMATICS

N. LASHNEV

ON CLOSED IMAGES OF METRIC SPACES

(Presented by Academician P. S. Aleksandrov, January 3, 1966)

In the present note* a necessary and sufficient condition is established for a \(T_1\)-space to be the image of a metric (complete metric) space under a closed continuous mapping (in what follows all mappings will be assumed continuous).

It follows from the Nagata—Smirnov theorem \((^{3,4})\) that metrizable spaces are characterized among all regular spaces by the existence in them of a refining sequence of locally finite covers. The simplest examples show that neither this property nor properties close to it are preserved under closed mappings, if one speaks of systems of open sets. The point is that under closed mappings the character of the space may increase at some points.

The situation is different with systems of closed sets. The property of a space to have a refining sequence of locally finite closed covers (which also characterizes metrizability), after a slight generalization, becomes a characteristic of closed images of metric spaces.

§ 1. Definition 1. A system \(\{F_\alpha\}\), \(\alpha \in A\), of closed sets of a space \(X\) is called hereditarily conservative if, for any set of indices \(A' \subseteq A\) and any system \(\{M_\alpha\}\), \(\alpha \in A'\), of closed sets of the space \(X\), \(M_\alpha \subseteq F_\alpha\), the set
\[ \bigcup_{\alpha \in A'} M_\alpha \]
is closed in \(X\).

It follows immediately from the definition that

Lemma 1. Every locally finite system of closed sets of a topological space is hereditarily conservative.

Lemma 2. The property of a system of closed sets of being hereditarily conservative is preserved under closed mappings.

The proof is carried out by a simple verification.

Lemma 3. If a sequence \(\{x_n\}\) of pairwise distinct points converges in a \(T_1\)-space \(X\), and if a system of sets \(\mathfrak A\) is hereditarily conservative, then there exists a natural number \(n^*\) such that the star of the system \(\mathfrak A\) with respect to the set \(\{x_n\}\), \(n > n^*\), consists of a finite number of sets.

Proof. If the contrary is assumed, then one can choose a subsequence of points \(\{x_{n_i}\}\) and a sequence of sets \(\{B_i\}\) of the system \(\mathfrak A\) such that \(x_{n_i} \in B_i\) and the sets \(B_i\) are pairwise distinct. Since the set \(\{x_{n_i}\}\) is not closed, we obtain a contradiction with the hereditary conservativeness of the system \(\mathfrak A\).

Definition 2. A sequence of closed covers \(\{\mathfrak A_i\}\) of a space \(X\) is called almost refining if, for every point \(x_0 \in X\), any system of sets \(\{B_i\}\), \(B_i \in \mathfrak A_i\), \(x_0 \in B_i\), either

* See also the author’s preceding note \((^8)\).

is hereditarily conservative, or forms a net* of the space at the point \(x_0\).

Theorem 1. In order that a \(T_1\)-space \(X\) be a closed image of a metric space, it is necessary and sufficient that the following conditions be satisfied simultaneously:

1) there exists in \(X\) an almost refining sequence of hereditarily conservative covers forming a net of the space \(X\);

2) \(X\) is a Fréchet–Urysohn space**.

Proof. To prove the necessity of the first condition, suppose that \(S\) is a metric space and that the mapping \(f:S\to X\) is closed. Taking in \(S\) such a sequence of locally finite closed covers \(\mathfrak A_i\) that the diameter of each \(F_\alpha^i\in\mathfrak A_i\) does not exceed \(1/i\), we obtain, by Lemmas 1 and 2, that the image \(f\mathfrak A_i\) of each cover \(\mathfrak A_i\) is a hereditarily conservative cover. Relying on the fact that the mesh of the covers \(\mathfrak A_i\) tends to zero as \(i\to\infty\), it is easy to verify that the covers \(f\mathfrak A_i\) form an almost refining sequence in \(X\).

Sufficiency. Let \(\{\mathfrak A_i\}\) be an almost refining sequence of hereditarily conservative covers forming a net of the space \(X\). We consider the product

\[ \prod_{i=1}^{\infty}\mathfrak A_i \]

of the systems \(\mathfrak A_i\) as a set lying in the Baire space \(B(\tau)\) of the corresponding weight***. The points of this set are all possible collections \(\{A_i\}\) of sets \(A_i\), one from each system \(\mathfrak A_i\). Denote by \(S\) the set of all such collections

\[ \{A_i\}\in\prod_{i=1}^{\infty}\mathfrak A_i, \]

each of which forms a net of some point \(x_0\in X\). Then \(S\) is a metrizable space. Assigning to a collection \(\{A_i\}\in S\) that unique point \(x_0\in X\) of which it is a net, we obtain a mapping \(f\) of the space \(S\) into the space \(X\).

\(1^\circ.\) \(f\) is a mapping onto the whole space \(X\). Let \(x_0\) be an arbitrary point of the space \(X\). If \(x_0\) is isolated, then, since

\[ \bigcup_{i=1}^{\infty}\mathfrak A_i \]

forms a net of the space \(X\), in some \(\mathfrak A_i\) there is a one-point element \(x_0\in\mathfrak A_i\). Any collection \(\{A_i\}\) containing \(x_0\) is mapped to \(x_0\). If, however, \(x_0\in [X\setminus x_0]\), then in \(X\setminus x_0\) there exists a sequence \(\{x_n\}\) converging to \(x_0\), since \(X\) is a Fréchet–Urysohn space. Using the conservativity of the systems \(\mathfrak A_i\), one can choose a sequence \(\{x_{n_i}\}\subseteq \{x_n\}\) and choose from each system \(\mathfrak A_i\) a set \(A_i\) so that \(x_{n_i}\in A_i\), \(x_0\in A_i\) for all \(i\). Since \(\{x_{n_i}\}\) is a nonclosed set in \(X\), the system \(\{A_i\}\) is not hereditarily conservative and, by the condition, forms a net of the point \(x_0\).

\(2^\circ.\) The mapping \(f\) is continuous. This follows immediately from the definition of the topology in the space \(S\) and from the fact that each collection \(\xi=\{A_i\}\in S\) forms a net of the point \(f\xi\) in \(X\).

\(3^\circ.\) The mapping \(f\) is closed. Let \(N\) be an arbitrary set in the space \(S\), and let \(M\) be its image in \(X\). Suppose that \(M\) is not closed in \(X\), and prove that \(N\) is not closed in \(S\).

There exists a point \(x_0\in [M]_X\setminus M\). Since \(X\) is a Fréchet–Urysohn space, there exists a sequence \(\{x_n\}\) of points of \(M\) converging to \(x_0\). In the preimage of each point \(x_n\) choose a point \(\xi_n\) belonging to the set \(N\). Denoting by \(O_{ix_0}\) the complement to the sum

* For the definition of the concept of a net, see (1).

** For the definition of the Fréchet–Urysohn property and the preservation of this property under closed mappings, see (2).

*** For the technique of constructing mappings of sets lying in Baire space, see the papers (6, 9).

of all sets of the system \(\mathfrak A_i\) not containing \(x_0\) (by virtue of the conservativity of the cover \(\mathfrak A_i\), the set \(O_i x_0\) is a neighborhood of the point \(x_0\)), we obtain that all \(x_n\), from some point on, lie in \(O_i x_0\). Using Lemma 3, one can find a natural number \(l_1\) such that the set of all elements of the system \(\mathfrak A_1\) intersecting \(\{x_n\}\), \(n>l_1\), is finite, and each of them contains \(x_0\). Therefore there exists an infinite subsequence \(\alpha_1\) of the sequence \(\{\xi_n\}\) such that every tuple \(\xi \in \alpha_1\) begins with one and the same set \(A_1 \in \mathfrak A_1\), containing \(x_0\). Continuing this process by induction, we find for every natural \(i\) an infinite set \(\alpha_i \subseteq \{\xi_n\} \subseteq S\) and a set \(A_i \in \mathfrak A_i\) such that \(\alpha_i \subseteq \alpha_{i+1}\), every tuple \(\xi \in \alpha_i\) has in the \(i\)-th place the set \(A_i\), and each \(A_i\) contains \(x_0\). Finally, in each \(\alpha_i\) choose a point \(\xi_{n_i}\) so that \(n_i<n_{i+1}\). By the last inequality, the sequence \(\{f\xi_{n_i}\}\) converges to the point \(x_0\). Since, moreover, for every \(i\) we have \(f\xi_{n_i}\in A_i\), the system \(\{A_i\}\) is not hereditarily conservative. Consequently, the tuple \(\xi_0=\{A_i\}\) forms a net of the point \(x_0\) in the space \(X\), i.e. \(\xi_0\in S\), \(f\xi_0=x_0\), and \(\xi_0\in N\). Obviously, the sequence \(\{\xi_{n_i}\}\) converges in the space \(S\) to the point \(\xi_0\), i.e. the set \(N\) is not closed in \(S\). The theorem is proved.

§ 2. A countable sequence of covers \(\{\mathfrak A_i\}\) of a topological space \(X\) is called an \(A\)-system,* if between certain elements of the covers \(\mathfrak A_i\) a subordination relation is defined in such a way that, for all \(i\), the following conditions are satisfied:

1. Each element \(B_{i+1}\in\mathfrak A_{i+1}\) is subordinate to one and only one element of the cover \(\mathfrak A_i\).

2. Each element \(B_i\in\mathfrak A_i\) is the union of all elements of the cover \(\mathfrak A_{i+1}\) subordinate to it.

A sequence \(\{B_i\}\) of sets of an \(A\)-system \(\{\mathfrak A_i\}\) is called a thread if, for every \(i\), \(B_{i+1}\) is subordinate to \(B_i\).

Theorem 2. In order that a \(T_1\)-space \(X\) be a closed image of a metrizable space with a complete metric, it is necessary and sufficient that the following two conditions be fulfilled simultaneously:

1) there exists in \(X\) an \(A\)-system consisting of hereditarily conservative covers, each thread of which forms a net of some point \(x\in X\);

2) \(X\) is a Fréchet–Urysohn space.

The methods of proof are the same as in Theorem 1; here the metric space \(\Xi\), whose image is our space \(X\), may be assumed to lie in Baire space as a closed subspace (cf. in this connection (7)).

The author expresses gratitude to V. I. Ponomarev, under whose supervision this work was written.

Moscow State University
named after M. V. Lomonosov

Received
22 XII 1965

REFERENCES

1. A. V. Arkhangel’skii, DAN, 126, No. 2 (1959).
2. A. V. Arkhangel’skii, DAN, 153, No. 4 (1963).
3. I. Nagata, J. Inst. Polytechn. Osaka City University, 1, 93 (1950).
4. Yu. M. Smirnov, UMN, 6, no. 6, 100 (1951).
5. G. Aleksandrov, V. Ponomarev, Fundam. Math., 50, No. 4, 449 (1962).
6. V. I. Ponomarev, Bull. Polish Acad. Sci., ser. math., phys., 8, 3, 127 (1960).
7. A. H. Stone, Rozprawy Matem., 28, 3 (1962).
8. N. Lashnev, DAN, 165, No. 4 (1965).
9. A. V. Arkhangel’skii, DAN, 145, No. 2, 245 (1962).

* \(A\)-systems of covers were defined in (5).