MATHEMATICS

M. SHERSHEV

STRONG DIMENSION OF MAPPINGS AND THE DIMENSIONAL CHARACTERISTIC ASSOCIATED WITH IT FOR ARBITRARY METRIC SPACES*

(Presented by Academician P. S. Aleksandrov on 11 VI 1960)

Let a mapping \(f\) of a space \(X\) into a space \(Y\) be given. Slightly modifying M. Katětov’s definition \((^{1})\), we shall call the mapping \(f\) strongly zero-dimensional , if for every number \(\varepsilon > 0\) there is a number \(\delta > 0\) such that, if the diameter of an open set \(O\) of the space \(Y\) is less than \(\delta\), then its full inverse image \(f^{-1}(O)\) is the sum of open pairwise disjoint sets \(H_\alpha\), each of which satisfies \(\operatorname{diam} H_\alpha < \varepsilon\). Following Yu. Smirnov, we shall call the strong dimension* of the mapping \(f\) the least of those natural numbers \(k\) such that the space \(X\) decomposes into the sum of \(k+1\) sets \(X_i\), on each of which the mapping \(f\) is strongly zero-dimensional. We shall agree to denote the strong dimension by \(\operatorname{Dim} f\).

The principal aim of this note is the following two theorems, which give the desired characteristic of the dimension \(\dim X\), one that is new even for spaces with a countable base:

Theorem 1. Let \(f\) be a mapping of the space \(X\) into the space \(Y\); then

\[ \dim X \leqslant \dim Y + \operatorname{Dim} f. \]

Theorem 2. If \(\dim X = n\), \(k \leqslant n\), then there exists a bounded** mapping \(f\) of the space \(X\) into the Euclidean space \(E^k\) such that \(\operatorname{Dim} f = n-k\); moreover, the set of such mappings is everywhere dense in the space \(C(X, E^k)\) of all bounded mappings of the space \(X\) into \(E^k\).

Corollary. The dimension of the space \(X\) is at most \(n\) if and only if, for some (for every!) \(k \leqslant n\), there exists a mapping \(f\) into \(E^k\) such that \(\operatorname{Dim} f \leqslant n-k\).

These theorems are a generalization of the known analogous theorems of W. Hurewicz \((^{3,4})\) for compacta, where the dimension of mappings \(\dim f\) is defined by means of the dimension of the full inverse images of points. For strongly zero-dimensional mappings these theorems were proved by M. Katětov \((^{1a})\). It is known that Hurewicz’s first theorem in its direct formulation is not true even for spaces with a countable base: discarding in the known example

* Only the dimension of a space defined by means of coverings is considered.

** By a space we everywhere mean only a metric space, and by a mapping only a continuous mapping.

*** M. Katětov \((^{1})\) called such mappings uniformly zero-dimensional. It seems to us correct to reserve this term for that concept of dimension of mappings which would give an analogous characteristic of uniform dimension in the sense of Yu. Smirnov \((^{2})\).

* A mapping \(f\) of a space \(X\) into a space \(Y\) is called *bounded if the image \(f(X)\) is a bounded set.

of a completely disconnected set of Knaster—Kuratowski \((^5)\) the vertex, we see that the projection \(\pi\) (from this point) of the remaining one-dimensional set onto the zero-dimensional Cantor set has dimension \(\dim \pi=0\). Moreover, Hurewicz \((^6)\) gave, for every \(n\), a general method of constructing \(n\)-dimensional completely disconnected sets \(X^n\), mapped one-to-one onto a Cantor set, and consequently also onto a line.

At the same time, the first Hurewicz theorem in its original form is valid for arbitrary spaces if only closed mappings are considered \((^7,^8)\). Nevertheless, P. S. Aleksandrov’s tempting hypothesis of obtaining a dimension characteristic by replacing, in Hurewicz’s theorems, arbitrary mappings by closed ones was refuted by A. H. Stone \((^9)\), who constructed an example of a plane one-dimensional set for which there exists no closed zero-dimensional mapping onto a line. We now turn to the exposition of the paper.

Theorem A. Let a mapping \(f\) of a space \(X\) into a space \(Y\) with a countable base be given; then, in the space \(X\), any of its subsets \(M\) can be enclosed in a set \(M_0\) of type \(G_\delta\), on which the mapping \(f\) has the same strong dimension as on \(M\): \(\operatorname{Dim}_{M_0} f=\operatorname{Dim}_{M} f\).

Proof. We may assume that \(\operatorname{Dim}_{M} f=0\). For each number \(\varepsilon_k=1/2k\) there exists a number \(\delta_k>0\) such that the condition of strong zero-dimensionality is fulfilled on \(M\). Consider the spherical neighborhoods \(O(c_i,\delta_k)\) of the points \(c_i\) of some countable dense subset \(C\) in \(Y\). Let
\[ \Gamma_{ik}=f^{-1}(O(c_i,\delta_k)) \]
and
\[ H_{ik}=M\cap \Gamma_{ik}. \]
There are decompositions
\[ H_{ik}=\bigcup_\alpha H_{ik}^{\alpha}, \]
where the \(H_{ik}^{\alpha}\) are open in \(M\),
\[ \operatorname{diam} H_{ik}^{\alpha}<\frac{1}{2k},\qquad H_{ik}^{\alpha}\cap H_{ik}^{\beta}=\varnothing \]
(if \(\alpha\ne\beta\)), where \(\varnothing\) is the empty set. For any fixed \(i\) and \(k\), by Yu. Smirnov’s lemma \((^{10})\), the sets \(H_{ik}^{\alpha}\) open in \(M\) can be extended to sets \(\Gamma_{ik}^{\alpha}\) open in \(X\) so that
\[ \operatorname{diam}\Gamma_{ik}^{\alpha}\leq \frac{1}{k},\qquad \Gamma_{ik}^{\alpha}\cap\Gamma_{ik}^{\beta}=\varnothing,\quad \text{if } \alpha\ne\beta, \]
and so that
\[ M\cap \Gamma_{ik}=\bigcup_\alpha \Gamma_{ik}^{\alpha}\subseteq \Gamma_{ik}. \]
The sets
\[ V_{ik}=\Gamma_{ik}\setminus\bigcup_\alpha \Gamma_{ik}^{\alpha}, \]
and together with them also the set
\[ V=\bigcup_{i,k}V_{ik} \]
are of type \(F_\sigma\). The set
\[ M_0=X\setminus V \]
is the desired one.

Theorem B*. The projection \(\pi\) of Euclidean space \(E^n\) onto its subspace \(E^k\) has strong dimension \(\operatorname{Dim}\pi=n-k\).

Proof. Decompose the space \(E^{\,n-k}\) into the sum of \(n-k+1\) zero-dimensional sets \(N_j\). On each of the sets \(E^k\times N_j\) the mapping \(\pi\) is strongly zero-dimensional, and
\[ E^n=\bigcup_j (E^k\times N_j). \]
The theorem is proved.

Lemma 1. If, in the superposition \(fg\) of two mappings, one of them is strongly zero-dimensional, then
\[ \operatorname{Dim} fg=\operatorname{Dim} f+\operatorname{Dim} g. \]

The proof is easily reduced to the simple case of strong zero-dimensionality of both mappings.

Lemma 2. For every mapping \(g\) of a space \(X\) into a space \(Y\) one can find a mapping \(\varphi\) of the space \(X\) into the space \(E^{\dim Y}\) such that
\[ \operatorname{Dim}\varphi=\operatorname{Dim}g. \]

Proof. By a theorem of M. Katětov \((^{16})\), there exists **a strongly zero-dimensional mapping \(f\) of the space \(Y\) into \(E^{\dim Y}\). The superposition \(fg\) is the required one.

* True, of course, is the following more general theorem:

Theorem B′. Let the product \(X\times Y\) of spaces \(X\) and \(Y\) be given; then for the projection \(\pi\) onto the space \(X\) we have \(\operatorname{Dim}\pi=\dim Y\).

The proof is carried out in exactly the same way with the aid of Katětov’s theorem \((^{1a})\) on the possibility of decomposing an \(n\)-dimensional space into the sum of \(n+1\) zero-dimensional sets.

** Even in the case when \(\dim Y=\infty\). By \(E^\infty\) one should, of course, understand Hilbert space.

Proof of Theorem 1. By the preceding, in the condition of the theorem one may replace* the space \(Y\) by \(E^n\), where \(n=\dim Y\). The space \(E^n\) is easily represented as the sum of \(n+1\) zero-dimensional sets \(N_i\) in such a way that every sum of the form \(\bigcup_{i=0}^{p} N_i\) is of type \(F_\sigma\). Let \(N'_i=f^{-1}(N_i)\), and let \(k=\operatorname{Dim} f\). By Theorem A, the space \(X\) can be represented as the sum of \(k+1\) sets \(X_j\) of type \(G_\delta\), on each of which \(\operatorname{Dim}_{X_j} f=0\). Put \(X'_j=X_j\setminus\bigcup_{i>j}X_i\). Then every sum of the form \(\bigcup_{j=0}^{q} X'_j\) is of type \(F_\sigma\). Finally, let \(H_{ij}=N'_i\cap X'_j\). Since \(\dim N_i=0\) and \(\operatorname{Dim} X'_j f=0\), each of the sets \(H_{ij}\) has a base decomposing into the sum of a countable number of open coverings of multiplicity 1. By Morita’s theorem \((^8)\), then \(\dim H_{ij}=0\) for all \(i\) and \(j\). One can show that under our assumptions every term \(H_{ij}\) of the sum \(D_l=\bigcup_{i+j=l} H_{ij}\) is in it of type \(F_\sigma\).

Therefore \(\dim D_l=0\). But \(X=\bigcup_{l=0}^{n+k}D_l\).

Hence, by a theorem of Yu. Smirnov \((^{11})\), \(\dim X\le n+k\), as was required to prove.

Proof of Theorem 2. Let \(k\le n=\dim X\). By a theorem of M. Katetov \((^1)\), there exists a strongly zero-dimensional mapping \(\varphi\) of the space \(X\) into \(E^n\) (even for \(n=\infty\))**. For the superposition \(\pi\varphi\), where \(\pi\) is the projection of the space \(E^n\) onto the subspace \(E^k\), by Theorem 1 we have \(\dim \pi\varphi\ge n-k\). At the same time \(\dim \pi\varphi\le n-k\) by Theorem A and Lemma 1. It is known \((^{12})\) that the set of all such superpositions is dense in \(C(X,E^k)\). The theorem is proved.

Remark. We have obtained a purely metric characteristic of the topological notion of dimension. Therefore, as a consequence, one can formulate a necessary condition for all metrics of the space \(X\), and a sufficient condition for some metric of the space \(X\).

Moscow State University
named after M. V. Lomonosov

Received
11 VI 1960

CITED LITERATURE

\(^1\) a) M. Katětov, DAN, 79, No. 2, 189 (1951); b) Czechoslovak Math. J., 2 (77), 333 (1952).
\(^2\) Yu. Smirnov, Matem. sborn., 38, No. 3, 283 (1956).
\(^3\) W. Hurewicz, Proc. Acad. Amsterdam, 30, 163 (1927).
\(^4\) W. Hurewicz, Sitzungsber. Preuss. Akad., 24, 754 (1933).
\(^5\) B. Knaster, K. Kuratowski, Fund. Math., 2, 206 (1921).
\(^6\) A. Hilgers, Fund. Math., 28, 303 (1937).
\(^7\) V. Gurevich, G. Vollmen, Dimension Theory, IL, 1948.
\(^8\) K. Morita, Proc. Japan Acad., 32, No. 3, 161 (1956).
\(^9\) A. Taimanov, UMN, 15, No. 5, 199 (1960).
\(^10\) Yu. Smirnov, Izv. AN SSSR, ser. matem., 20, 253 (1956).
\(^11\) Yu. Smirnov, Matem. sborn., 29, No. 1, 157 (1951).
\(^12\) M. Shershnev, UMN, 12, No. 5, 251 (1957).

* This is necessary for the time being, since the space \(Y\) need not possess a countable base.

** In this case Theorem 2 must be formulated as follows:

Theorem 2′. If \(k\le n=\dim X\), then there exists a bounded mapping \(f\) of the space \(X\) into the space \(E^{\,n-k}\) such that \(\operatorname{Dim} f=k\); moreover, the set of all such mappings is dense in the space \(C(X,E^{\,n-k})\) (\(\infty-k=\infty\) for every finite \(k\). \(\infty-\infty\) may mean any natural number and even \(\infty\)).