MATHEMATICS

A. ZARELUA

ON HUREWICZ’S THEOREM

(Presented by Academician P. S. Aleksandrov on 3 July 1961)

In § 1 the sum theorem is extended to point-finite open covers*, and Hurewicz’s theorem on lowering dimension under mappings is proved in Morita’s form (¹)
\(\dim X \leq \operatorname{Ind} Y + \dim f\)** under the assumption of weak paracompactness*** of the image. In § 2 the notions of a fully paracompact space and of a fully paracompact decomposition are introduced and studied. For metrizable spaces the first notion coincides with the notion of strong metrizability—the existence of a base decomposable into the sum of a countable number of star-finite**** covers (apparently introduced by Morita). An example is given of a strongly metrizable space which is not decomposable into the sum of a countable number of closed strongly paracompact***** spaces. In § 3 the same Hurewicz theorem is proved in the form of Yu. Smirnov (²)
\(\dim X \leq \operatorname{ind} Y + \dim f\) for fully paracompact decompositions, in particular for fully paracompact spaces. For these spaces the inequality \(\dim X \leq \operatorname{ind} X\) follows from this.

§ 1. Theorem 1. If a normal space \(R\) has a point-finite open cover \(\omega\) such that \(\dim U \leq n\) whenever \(U \in \omega\), then \(\dim R \leq n\).

Lemma 1. For a point-finite open cover \(\omega\), for each \(k=1,2,\ldots\) the set \(T_k\), consisting of all those points each of which belongs to no more than \(k\) elements of the cover \(\omega\), is closed; for every neighborhood \(O\) of the set \(T_{k-1}\), the difference \(T_k \setminus O\) is the body****** of a closed discrete******* system inscribed in \(\omega\).

For the proof of the theorem we prove by induction that \(\dim T_k \leq n\). Indeed, if \(\Phi \subseteq T_k \setminus T_{k-1}\) and \(\Phi\) is closed, then, by the lemma, \(\dim \Phi \leq n\). Hence, by Dowker’s lemma (³), we have \(\dim T_k \leq n\).

Corollary. If the space \(R\) is normal and weakly paracompact, then \(\operatorname{loc}\dim R = \dim R\)********.

* A cover is called point-finite if each point is contained in only finitely many elements of this cover.

** The dimension \(\dim\) is the dimension defined by means of covers; \(\operatorname{Ind}\) is the large inductive dimension, \(\operatorname{ind}\) the small inductive dimension;
\(\dim f = \sup_{y \in Y} \dim f^{-1}y\).

*** A space is weakly paracompact if every one of its open covers has an inscribed point-finite open cover.

**** A cover is called star-finite if each of its elements intersects only finitely many other elements.

***** A space is strongly paracompact if every one of its open covers has an inscribed open star-finite cover.

****** The body of a system is the union of all elements of this system.

******* A system of sets \(A_\lambda\) is discrete if every point \(x \in R\) has a neighborhood intersecting no more than one set \(A_\lambda\).

******** The local dimension \(\operatorname{loc}\dim R\) of the space \(R\) is the least of those numbers \(n\) such that every point \(x \in R\) has a neighborhood \(Ox\) such that \(\dim [Ox] \leq n\) (see (³)).

Theorem 2. Let \(g\) be a closed* mapping of a normal space \(X\) onto a weakly paracompact normal space \(Y\); then
\[ \dim X \leq \operatorname{Ind} Y+\dim g . \]

Proof. Suppose that, in the case \(\operatorname{Ind} Y' \leq m-1\), the theorem is true, and let \(\operatorname{Ind} Y=m\). Let \(\Phi\) be closed in \(X\), and let \(f\) be a mapping of the set \(\Phi\) into the sphere \(S^{n}\), \(n=m+\dim g\). Since \(\dim g^{-1}(y)\leq n\), \(f\) can be extended to \(\Phi\cup g^{-1}y\) and, hence, to some neighborhood \(V_y\) of it. The sets
\[ O_y=Y\setminus g(X\setminus V_y) \]
form an open cover of the space \(Y\). We inscribe in it a locally finite cover \(\omega\). Let \(S_k=g^{-1}(T_k)\). By induction one can prove that there exist open sets \(\Gamma_k\) satisfying the following conditions: a)
\[ \bigcup_{k\leq l} S_k \subseteq \bigcup_{k\leq l}\Gamma_k \]
for every \(l\); b) \(\dim \operatorname{Fr}\Gamma_k\leq n-1\); c) \(G_k=g(\Gamma_k)\) is open in \(Y\); d) \(g^{-1}G_k=\Gamma_k\); e) the mapping \(f\) is extended to \(\Phi\cup[\Gamma_k]\). By Hurewicz’s lemma, generalized by Yu. Smirnov in \((^{2})\), the mapping \(f\) is extended to all of \(X\). Hence \(\dim X\leq n\), as was required to prove.

1. We shall say that a cover \(\beta\) is weakly inscribed in a cover \(\alpha\) if one can choose from \(\beta\) a subcover inscribed in \(\alpha\).** We shall call a regular space fully paracompact if, in every one of its open covers, one can weakly inscribe an open cover decomposing into the sum of a countable number of star-finite covers.

Theorem 3. Every fully paracompact space is paracompact; every strongly paracompact space is fully paracompact.

For the proof of the first assertion, in view of a well-known theorem of Michael \((^{4})\), only the following is needed.

Lemma 2. In every open cover of a fully paracompact space one can inscribe an open cover decomposing into the sum of a countable number of discrete subsystems.

The lemma follows from the fact that every open star-finite cover \(\omega_i\) decomposes into the sum of countable or finite subsystems \(\omega_{i\lambda}\), whose bodies are pairwise disjoint and are open-closed sets \((^{5})\). Taking one element from each subsystem \(\omega_{i\lambda}\), we obtain a discrete system. Therefore the sum of the star-finite covers \(\omega_i\) also decomposes into the sum of a countable number of discrete subsystems.

We shall call a regular space strongly metrizable if it has a base decomposing into the sum of a countable number of star-finite covers.***

Lemma 3. A metrizable space is strongly metrizable if and only if it is fully paracompact.

It is easy to see that the product of a countable number of strongly metrizable spaces is strongly metrizable and that subspaces of a strongly metrizable space are also strongly metrizable. For strongly paracompact spaces these properties do not hold \((^{6})\).

Example 1. A strongly metrizable space \(S\) that is not the sum of a countable number of closed strongly paracompact sets.

Let \(\prod I_k\) be the topological product of a countable number of (open) intervals \(I_k\), and let \(B^\tau\) be the Baire space of uncountable weight \((^{5})\tau\). Then
\[ S=B^\tau\times \prod I_k . \]
One can show that every neighborhood of the space \(S\) contains a closed subset of \(S\) that is not strongly paracompact. Hence every closed strongly paracompact set is nowhere dense in \(S\). From this everything follows by the completeness of \(S\) and Baire’s theorem.

* A continuous mapping is called closed if the image of every closed set is closed.

** Every base of open sets is weakly inscribed in every open cover of the given space.

*** By the metrization theorem of Nagata—Smirnov \((^{7})\), every strongly metrizable space is metrizable.

Theorem 4. Every set of type \(F_\sigma\) in a completely paracompact space is completely paracompact.

Example 2. A closed mapping that maps a strongly paracompact metrizable space into a space that is not completely paracompact.

Take an uncountable number of discretely arranged segments of length 1 and identify the initial points of these segments into one point, so as to obtain a “nonmetrizable hedgehog.”

Note that, by mapping a “metrizable hedgehog” to one point, we obtain a closed mapping of a metrizable, not strongly paracompact space onto a compactum.

By a decomposition of a space \(R\) we shall mean a system of pairwise disjoint closed sets whose sum is equal to \(R\). A covering \(\gamma\) will be called a covering of the decomposition \(\beta\) if \(\beta\) is inscribed in \(\gamma\). A decomposition \(\beta\) of a space \(R\) will be called completely paracompact* if in every open covering of this decomposition one can weakly inscribe an open covering of the space \(R\) which decomposes into the sum of a countable number of star-finite coverings of the space \(R\).

Lemma 4. Let \(g\) be a closed mapping of a space \(X\) onto a space \(Y\); if one of these spaces is completely paracompact, then the decomposition generated by the mapping \(g\) is also completely paracompact.

In the case of complete paracompactness of the space \(X\), the closedness of the mapping is not needed.

Example 3. A closed mapping of a space that is not completely paracompact onto a space that is not completely paracompact, generating a completely paracompact decomposition.

Let \(X\) be the sum of an uncountable number of discretely arranged “metrizable hedgehogs.” In each of them choose one segment \(A_\lambda O_\lambda\), where \(O_\lambda\) is the center of the hedgehog. Map each segment \(A_\lambda O_\lambda\) isometrically onto the corresponding segment \(A'_\lambda O'\) of the “nonmetrizable hedgehog” \(Y\). Map all remaining points of the space \(X\) to the center \(O'\) of the hedgehog \(Y\).

p. 3. Theorem 5. Let in a normal space \(R\), for each neighborhood \(O\) of each element \(F\) of a decomposition \(\beta\), there exist a neighborhood \(V\) such that \(F \subset V \subset O\) and such that \(\dim \operatorname{Fr} V \leq n - 1\); then, if \(\beta\) is completely paracompact and if \(\dim F \leq n\) for every \(F \in \beta\), then also \(\dim R \leq n\).

Lemma 5. Let an open covering \(\gamma\) weakly contain an open covering decomposing into the sum of a countable number of star-finite coverings; then in \(\gamma\) one can inscribe an open covering, decomposing into the sum of a countable number of discrete systems, such that the boundary of each of its elements will lie in the boundary of some element of the covering \(\gamma\).

Proof of the theorem. Let \(\beta\) be a completely paracompact decomposition, and let \(\Phi\) be a closed set of the space \(R\). Let \(f\) be a mapping of the set \(\Phi\) into the sphere \(S^n\). Just as before, \(f\) can be extended to some neighborhood \(O_F\) of each set \(F \in \beta\). We may assume that \(\dim \operatorname{Fr} O_F \leq n - 1\). By Lemma 5 one can find such open sets \(\Gamma_i\) that \(\dim \operatorname{Fr} \Gamma_i \leq n - 1\) and such that \(f\) is extendable to \(\Phi \cup [\Gamma_i]\) for every \(i = 1, 2, \ldots\). Hence, \(f\) is extendable to all of \(R\), and \(\dim R \leq n\). The theorem is proved.

Theorem 6. Let \(g\) be a closed mapping of a normal space \(X\) onto a normal space \(Y\); if the mapping \(g\) generates a completely paracompact decomposition of the space \(X\), then \(\dim X \leq \operatorname{ind} Y + \dim g\).

Lemma 6. Let \(\beta=\{F\}\) be a decomposition of a space \(R\), and let \(\beta'\) be such a subsystem of it that the sum

\[ A=\bigcup_{F\in\beta'} F \]

is closed; then complete para-

* Every continuous mapping \(g\) generates a decomposition of the preimage \(X\) into the full preimages \(g^{-1}(y)\) of points \(y \in Y\).

compactness of the decomposition \(\beta\) entails the complete paracompactness of the decomposition \(\beta'\) on \(A\).

Proof of the theorem. Suppose that in the case \(\operatorname{ind} Y' < k\) the theorem is true, and let \(\operatorname{ind} Y = k\). Apply Theorem 5. For an arbitrary neighborhood \(O\) of the inverse image \(g^{-1}(y)\), by virtue of the closedness of the mapping \(g\) there exists a neighborhood \(Oy\) of the point \(y\) such that \(g^{-1}(Oy) \subseteq O\) and \(\operatorname{ind} \operatorname{Fr} Oy \leq k-1\). Lemma 6 together with the induction hypothesis leads to the inequality
\(\dim g^{-1}(\operatorname{Fr} Oy) \leq \dim g + k - 1\), and hence also to the inequality
\(\dim \operatorname{Fr} g^{-1}(Oy) \leq \dim g + k - 1\), which was required to be proved.

Corollary 1. If \(g\) is a closed mapping of a normal space \(X\) onto a normal space \(Y\), then the inequality
\(\dim X \leq \dim g + \operatorname{ind} Y\) holds in the following two cases: 1) either \(X\) or \(Y\) is the sum of a countable number of closed completely paracompact sets; 2) or \(X\) or \(Y\) is the sum of completely paracompact sets forming a locally finite system, all of which, except possibly one, are closed.

This is derived from Lemma 4 by Dowker’s method from \((^3)\).

Corollary 2. For every normal space \(R\) satisfying one of the conditions of the preceding corollary, in particular for a completely paracompact space, one always has
\(\dim R \leq \operatorname{ind} R\).

Hence, from the well-known theorem of Katetov—Morita \((^8)\), we have:

Corollary 3. For every metrizable space \(R\) that decomposes into the sum of a countable number of completely paracompact (strongly metrizable) closed sets, the fundamental dimensions are equal:
\(\dim R = \operatorname{ind} R = \operatorname{Ind} R\).

Theorem 7. If in a completely paracompact space \(R\) the sum theorem holds for the dimension \(\operatorname{Ind}\), then
\(\operatorname{Ind} R = \operatorname{ind} R\).

The proof is based on the following lemma:

Lemma 7. Let \(R\) be a completely paracompact space of dimension \(\operatorname{ind} R \leq n\); then for every closed set \(A\) and every neighborhood \(OA\) of it there exists a neighborhood \(UA\) such that \(UA \subseteq OA\) and such that the boundary \(\operatorname{Fr} UA\) decomposes into the sum of a countable number of closed sets \(C_i\) of dimension \(\operatorname{ind} C_i \leq n-1\).

Remark. The corollary of Theorem 1 and Theorem 2 are true under the following assumptions: 1) the space \(R(Y)\) is the sum of a countable number of closed weakly paracompact sets; 2) \(R(Y)\) is the body of a locally finite system of weakly paracompact sets, all of which are closed except, possibly, one.

I express my deep gratitude to Prof. Yu. M. Smirnov for his help in preparing this article. I note that all new definitions and Examples 2 and 3 belong to Yu. M. Smirnov.

Moscow State University
named after M. V. Lomonosov

Received
3 VII 1961

CITED LITERATURE

\(^1\) K. Morita, Proc. Japan Acad., 32, No. 3, 161 (1956).
\(^2\) Yu. M. Smirnov, Matem. sborn., 29 (71), No. 1, 157 (1951).
\(^3\) C. H. Dowker, Quart. J. Math., 6, No. 22, 101 (1955).
\(^4\) E. Michael, Proc. Am. Math. Soc., 4, No. 3, 831 (1953).
\(^5\) Yu. M. Smirnov, Izv. AN SSSR, ser. matem., 20, 253 (1956).
\(^6\) J. Nagata, J. Inst. Pol. Osaka City Univ., 8, No. 1, Ser. A, 9 (1957).
\(^7\) Yu. M. Smirnov, Uspekhi Mat. Nauk, 6, No. 6, 100 (1951).
\(^8\) M. Katetov, DAN, 79, No. 1, 189 (1951).