B. LEVSHENKO

ON INFINITE-DIMENSIONAL SPACES

(Presented by Academician P. S. Aleksandrov on 18 II 1961)

The first part of the note is devoted to a generalization of Hurewicz’s theorem on mappings that lower dimension (see (¹), p. 127) to weakly infinite-dimensional spaces*. In the second part of the note there are two examples. The first generalizes an example of a compactum, constructed by Yu. M. Smirnov (²), which is not decomposable into the sum of a countable number of closed finite-dimensional sets, but which has transfinite dimension and, consequently, is decomposable into the sum of a countable number of zero-dimensional sets. The second shows that even for compacta the sum theorem is no longer true with only two summands. Along the way it is proved that for the compacta \(Q^\alpha\) of Yu. M. Smirnov (²) the small** transfinite dimension assumes arbitrarily large values.

I. We shall call a space \(R\) weakly infinite-dimensional in the sense of P. S. Aleksandrov and in this case shall write \(\operatorname{Dim} R < \aleph\), if for every countable system of pairs of closed sets \(A_i, B_i\) such that \(A_i \cap B_i = \varnothing\) for all \(i\), there exist closed partitions*** \(C_i\) between \(A_i\) and \(B_i\) with empty intersection: \(\bigcap C_i = \varnothing\).

Theorem 1. If a bicompactum \(X\) is the sum of closed sets \(\Phi_\alpha\) of dimension \(\operatorname{Dim} \Phi_\alpha < \aleph\) and such that for every neighborhood \(O\Phi_\alpha\) of any set \(\Phi_\alpha\) there is a smaller neighborhood \(U\Phi_\alpha\) with boundary \(|U\Phi_\alpha|\) of dimension \(\operatorname{Dim} |U\Phi_\alpha| < \aleph\), then \(\operatorname{Dim} X < \aleph\).

Theorem 2. If a bicompactum \(X\) is mapped onto a bicompactum \(Y\) possessing small transfinite dimension, in such a way that \(\operatorname{Dim} f^{-1}y < \aleph\) for every point \(y \in Y\), then \(\operatorname{Dim} X < \aleph\).

Corollary. Every bicompactum possessing small transfinite dimension is weakly infinite-dimensional.

These basic results admit further generalizations. Preliminarily, let us call a space \(R\) weakly infinite-dimensional in the sense of Yu. M. Smirnov and in this case write \(\dim R < \aleph\), if in the preceding definition we require that there be partitions \(C_i\) with empty finite intersection: \(\bigcap_{i<n} C_i = \varnothing\), where \(n\) depends on the system of pairs \(\{A_i, B_i\}\).

Theorem 1′. If a strongly paracompact**** space \(X\) is the sum of closed sets \(\Phi_\alpha\) of dimension \(\dim \Phi_\alpha < \aleph\) and such—

* Spaces are everywhere assumed to be regular, and mappings continuous.

** Thus, there exist compacta of arbitrarily large transfinite dimension—a fact considered known but not found in the literature. Yu. M. Smirnov proved (²) that for the large transfinite dimension introduced by him \(\operatorname{Ind} Q^\alpha = \alpha\). The equality \(\operatorname{ind} R = \operatorname{Ind} R\) has not yet been proved even for compacta.

*** A partition in a set \(S\) between sets \(A\) and \(B\) is any set \(C\) such that \(S \setminus C = G \cup H\), where \(G\) and \(H\) are disjoint sets open in \(S\), with \(A \cap S \subseteq G\) and \(B \cap S \subseteq H\).

**** A space is strongly paracompact if into every open cover of it one can inscribe a star-finite cover (³), i.e. one such that each of its elements intersects only finitely many other elements. Every space with a countable base and even every finally compact (Lindelöf) space is strongly paracompact.

such that for every neighborhood \(O\Phi_\alpha\) of any set \(\Phi_\alpha\) there is a smaller neighborhood \(U\Phi_\alpha\) with boundary \(|U\Phi_\alpha|\) of dimension \(\operatorname{Dim}|U\Phi_\alpha|<\aleph\), then also \(\operatorname{Dim} X<\aleph\).

Theorem 2′. If \(f\) is a closed continuous mapping of a strongly paracompact space \(X\) onto a space \(Y\) representable as the sum of a countable number of subspaces having transfinite dimension, and if \(\dim f^{-1}(y)<\aleph\) for every point \(y\in Y\), then \(\operatorname{Dim} X<\aleph\).

Corollary. Every strongly paracompact space having small transfinite dimension is weakly finite-dimensional.

Proof of Theorem 1′. Let \(\{A^i,B^i; A_*^i,B_*^i\}\) be a countable system of pairs such that
\[ A^i\cap B^i=A_*^i\cap B_*^i=\varnothing . \]
Since \(\dim\Phi_\alpha<\aleph\), in \(\Phi_\alpha\) there exist partitions \(C_\alpha^i\) between \(A^i\) and \(B^i\) such that
\[ \bigcap_{i<n(\alpha)} C_\alpha^i=\varnothing . \]
Since \(X\) is normal,* there exist neighborhoods \(U\Phi_\alpha\) and partitions \(\bar C_\alpha^i\) lying in \([U\Phi_\alpha]\), between \(A^i\) and \(B^i\), such that \(\operatorname{Dim}|U\Phi_\alpha|<\aleph\) and
\[ \bigcap_{i<n(\alpha)} \bar C_\alpha^i=\varnothing . \]
Into the covering \(\{U\Phi_\alpha\}\) we inscribe a star-finite covering \(\omega\). It splits into subsystems \(\omega_\lambda\) such that each of them is countable, and the bodies
\[ R_\lambda=\widetilde{\omega}_\lambda \]
are open-closed and pairwise disjoint \((^3)\). Let \(\omega_\lambda=\{U_{\lambda j}\}\) and let \(a(\lambda j)\) be such that \(U_{\lambda j}\subset U\Phi_{a(\lambda j)}\). Let
\[ V_{\lambda j}=R_\lambda\cap \Phi U_{a(\lambda j)}, \]
and
\[ \bar C_{\lambda j}^i=R_\lambda\cap \bar C_{a(\lambda j)}^i,\qquad i\le n(a)=n(\lambda j). \]
We carry out the proof inside some piece \(R=R_\lambda\), omitting the index \(\lambda\). Suppose that in
\[ S_m=\bigcup_{j\le m}|V_j| \]
partitions \(D_m^i\), where \(i\le n(m)\), have been constructed between \(A^i\) and \(B^i\) such that
\[ \bigcap_{i\le n(m)}D_m^i\subset \bigcup_{i\le m-1}|S_i| \]
and suppose
\[ S_m\setminus D_m^i=G_m^i\cup H_m^i, \]
where \(G_m^i\) and \(H_m^i\) are disjoint open neighborhoods in \(S_m\) of the sets \(A^i\cap S_m\) and \(B^i\cap S_m\). Similarly,
\[ |V_{m+1}|\setminus \bar C_{m+1}^i=G_{m+1}^i\cup H_{m+1}^i, \]
where
\[ G_{m+1}^i\cap H_{m+1}^i=\varnothing,\qquad A^i\cap |V_{m+1}|\subset G_{m+1}^i,\qquad B^i\cap |V_{m+1}|\subset H_{m+1}^i . \]
The partitions \(D_{m+1}^i\) in \(S_{m+1}\), for \(i\le n(m)\), are obtained as follows:
\[ D_{m+1}^i = D_m^i \cup (S_{m+1}\cap \bar C_{m+1}^i\setminus S_m) \cup \bigl(|S_m|\cap |V_{m+1}|\setminus (G_{m+1}^i\cup H_{m+1}^i)\bigr), \]
where
\[ G_{m+1}^i = S_{m+1}\setminus (S_m\setminus G_m^i)\setminus (|V_{m+1}|\setminus G_{m+1}^i), \]
\[ H_{m+1}^i = S_{m+1}\setminus (S_m\setminus G_m^i)\setminus (|V_{m+1}|\setminus H_{m+1}^i). \]
For \(n(m)<i\le n(m+1)\), the partitions \(D_{m+1}^i\) are arbitrary closed partitions in \(S_{m+1}\) between \(A^i\) and \(B^i\) such that
\[ D_{m+1}^i\cap |V_{m+1}|=\bar C_{m+1}^i . \]
The sum
\[ D^i=\bigcup_m D_m^i \]
is a partition in the piece \(R=R_\lambda\) between \(A^i\) and \(B^i\), and
\[ \bigcap_i D^i\subset \bigcup_m |S_m|\subset \bigcup_j |V_j| . \]
Since \(\operatorname{Dim}V_j<\aleph\), it follows by \((^4)\) that
\[ \operatorname{Dim}\bigcap_i D^i<\aleph, \]
and hence, in \(\bigcap_i D^i\), there exist partitions \(C_*^i\) between \(A_*^i\) and \(B_*^i\) such that
\[ \bigcap_i C_*^i=\varnothing . \]
Then in \(R_\lambda=R\) there exist partitions \(D_*^i\) between \(A_*^i\) and \(B_*^i\) such that
\[ D_*^i\cap \bigcap_i D^i=C_*^i . \]
Consequently,
\[ \bigcap_i D_*^i\cap \bigcap_i D^i=\varnothing \]
and the pairs \((A^i,B^i)\), \((A_*^i,B_*^i)\) are separated in the piece \(R_\lambda\) as required. The theorem is proved.

Proof of Theorem 2′. Let \(X_i=f^{-1}Y_i\) and \(\operatorname{ind}Y_i\le \alpha_i\). We have:
\[ X_i=\bigcup_{y\in Y_i} f^{-1}y, \]
where \(\dim f^{-1}y<\aleph\), and (by the closedness of the mapping) into any neighborhood \(Of^{-1}y\) of each set \(f^{-1}y\) one can insert a neighborhood \(U=Uf^{-1}y\), for which
\[ \operatorname{ind} f|U|<\alpha_i . \]
By means of induction and Theorem 1′ we obtain \(\operatorname{Dim}X_i<\aleph\). But then \((^4)\) also gives \(\operatorname{Dim}X<\aleph\). The theorem is proved.

* By virtue of the normality of the regular strongly paracompact space \(X\), see \((^3)\).

Theorems 1 and 2 follow from theorems \(1'\) and \(2'\) because \(\dim R=\operatorname{Dim} R\) for bicompacts \((^4)\).

Corollary. The Tikhonov product of a weakly infinite-dimensional bicompactum by a space which is the sum of a countable number of closed sets having small transfinite dimension is weakly infinite-dimensional in the sense of Alexandrov.

II. Theorem 3. There exist compacta of every countable small transfinite dimension.

These are the compacta \(Q^\beta\) of Yu. M. Smirnov \((^2)\), defined as follows: by \(Q^n\) we denote the \(n\)-dimensional cube; for a limit index \(\beta\), the compactum \(Q^\beta\) is the Alexandrov compactification of the discrete sum \(\bigcup_{\alpha<\beta} Q^\alpha\); for an isolated index \(\beta\) we put
\[ Q^\beta=Q^{\beta-1}\times Q^1 . \]
The proof is by induction with a complicated inductive hypothesis, using the sum theorem for a finite number of spaces having transfinite dimension.

Theorem \(3'\). There exist compacta of every countable small transfinite dimension \(\alpha\) which cannot be represented as the sum of a countable number of subcompacta of smaller dimension.

Proof. Let \(I^n\) be the cube of the Hilbert parallelepiped \(I^\omega\), at whose points all coordinates \(y_i=0\) for \(i>n\). Let \(\Phi_1\) be a compactum homeomorphic to \(Q^\alpha\), lying in \(I^\omega\times I^1\). Next, by induction we construct compacta \(\Phi_i\) having the following properties: a) \(\Phi_i\subset I^\omega\times I^i\) and \(\operatorname{ind}\Phi_i=\alpha\); b) \(\Phi_i\subset\Phi_{i+1}\), \(\Phi_i\) is nowhere dense in \(\Phi_{i+1}\); c) if
\[ (x;y_1,\ldots,y_i,0,0,\ldots)\in\Phi_i, \]
then
\[ (x,y_1,\ldots,y_i,0,y_{i+2},0,\ldots)\notin\Phi_{i+2} \]
for \(y_{i+2}>0\), where \(x\in I^\omega\),
\[ (y_1,y_2,\ldots,y_i,0,\ldots)\in I^i, \]
and
\[ (y_1,\ldots,y_i,0,y_{i+2},0,\ldots)\in I^{i+2}. \]
If the compacta \(\Phi_1,\ldots,\Phi_n\) have been constructed, then the compactum \(\Phi_{n+1}\) is constructed in the following way. For the numbers
\[ \varepsilon_i=1/3^{\,i(n+1)} \]
we take finite \(\varepsilon_i\)-nets
\[ x_1^i,\ldots,x_{N(i)}^i \]
of the compactum \(\Phi_n\), such that \(x_j^i\notin\Phi_n\),
\[ \rho(x_j^i,\Phi_{n-1})>\varepsilon_i/2 \]
and
\[ \rho(x_{j_1}^i,x_{j_2}^i)>\varepsilon_i . \]
In the hyperplane \(I^\omega\times I^n\times\varepsilon_i\), lying in \(I^\omega\times I^{n+1}\), consider the balls \(U_j^i\) of radius \(\varepsilon_i/3\) with centers at the points \(x_j^i\) \((j\le N(i),\ i=1,2,\ldots)\), and in the ball \(U_j^i\) take a compactum \(\Phi_{n+1}^{ij}\), homeomorphic to \(Q^\alpha\) and such that
\[ \Phi_{n+1}^{ij}\subseteq (I^\omega\times I^n\times\varepsilon_i)\setminus (I^\omega\times I^{n-1}\times O\times\varepsilon_i). \]
By construction,
\[ \Phi_{n+1}=\overline{\bigcup_{i,j}\Phi_{n+1}^{ij}}. \]
Let
\[ E=\bigcup\Phi_n . \]
1) \(\operatorname{ind}E=\alpha\); 2) \(E\) is semicompact*; 3) every open subset of \(E\) has dimension \(\alpha\). By a theorem of E. G. Sklyarenko \((^5)\), in view of 2), there exists a compactum \(F^\alpha\), which is an extension of the set \(E\), such that
\[ \operatorname{ind}(F^\alpha\setminus E)=0. \]
Slightly modifying E. G. Sklyarenko’s construction, in view of the precautions taken by us, one can arrange that
\[ \operatorname{ind}F^\alpha=\alpha. \]
Since no compactum \(H\) of dimension \(<\alpha\) from \(F^\alpha\) is anywhere dense in \(F^\alpha\) (in view of 3)), \(F^\alpha\) is not the sum of a countable number of subcompacta of dimension \(<\alpha\).

Corollary. There exist spaces with a countable base which are the sum of a countable number of zero-dimensional sets, but are not the sum of a countable number of closed sets having transfinite dimension.

An example of such a set is Nagata’s universal space \((^6)\).

Example. Let us construct a compactum which is the sum of two subcompacta of smaller transfinite dimension than itself. This is the compactum \(Q^{\omega+1}\). By the preceding and by theorem 5 from \((^2)\) we have:
\[ \operatorname{ind}Q^{\omega+1}=\operatorname{Ind}Q^{\omega+1}= \]

* There is a base of open sets with compact boundaries.

\[ = \omega_0+1. \]
Let*
\[ I_n=\bigcup_{i=0}^{2^n-1}\left(\frac{2i}{2^{n+1}},\frac{2i+1}{2^{n+1}}\right),\qquad J_n=\bigcup_{i=0}^{2^n-1}\left(\frac{2i+1}{2^{n+1}},\frac{2i+2}{2^{n+1}}\right), \]
\[ A=\left[\bigcup_n\left(Q^n\times I_n\right)\right]\quad\text{and}\quad B=\left[\bigcup_n\left(Q^n\times J_n\right)\right]. \]
It is easy to see that \(\operatorname{ind} A\geqslant \omega_0\); \(A\) and \(B\) are homeomorphic; \(A\cup B=Q^{\omega_0+1}\). We shall show that \(\operatorname{Ind} A\leqslant \omega_0\). Let \(\Phi\) be closed in \(A\), and let \(U\) be a neighborhood, and \(\Phi'=\Phi\cap(\xi\times Q^1)\). There exists in \(Q^{\omega_0+1}\) a neighborhood \(V\times G\) of the set \(\Phi'\) such that \(V\subseteq Q^{\omega_0}\) and \(|V|=\varnothing\), while \(G\) is the sum of a finite number of intervals \((a_i,b_i)\) such that
\[ Q^{\omega_0}\times a_i\subseteq B\setminus A\cup(\xi\times Q^1)=\dot B \]
and \(Q^{\omega_0}\times b_i\subseteq \dot B\), and, moreover,
\[ [A\cap(V\times G)]\subseteq U. \]
Since
\[ \Phi\setminus(V\times G)\subseteq \bigcup_N Q^i, \]
there exists a neighborhood \(W\subseteq U\) of the set \(\Phi\setminus(V\times G)\) with finite-dimensional boundary. Then
\[ \Phi\subseteq W\cup(V\times G)\subseteq [W\cup(V\times G)]\subseteq U, \]
and \(W\cup(V\times G)\) has finite-dimensional boundary, as was required to be proved.

I express my heartfelt gratitude to Yu. M. Smirnov for the help and support he gave me in my work.

Moscow State University
named after M. V. Lomonosov

Received
31 XII 1960

References

1. V. Gurevich, G. Wallman, Dimension Theory, Moscow, 1948.
2. Yu. M. Smirnov, Izv. Acad. Sci. USSR, ser. math., 23, 185 (1959).
3. Yu. M. Smirnov, Izv. Acad. Sci. USSR, ser. math., 20, 253 (1956).
4. B. T. Levshenko, Vestn. Moscow Univ., No. 5, 219 (1959).
5. E. G. Sklyarenko, DAN, 120, No. 6, 1200 (1958).
6. J. Nagata, Func. Math., 28, 1 (1960).

* \((\ )\) denotes an interval on the line; \([\ ]\) denotes closure.