Abstract
Full Text
Reports of the Academy of Sciences of the USSR
1963. Volume 151, No. 4
MATHEMATICS
O. V. LOKUTSIEVSKII
ON A PROBLEM OF P. S. URYSOHN
(Presented by Academician P. S. Aleksandrov on 13 II 1963)
- The relative distance \(\rho^*(x_1,x_2)\) between points \(x_1,x_2\) of a continuum \(X\) is defined as the lower bound of the diameters of those of its subcontinua which contain both these points.* The function \(\rho^*(x_1,x_2)\) transforms \(X\) into a new metric space \(X^*\) (generally speaking, without a countable base), which is called the space of relative distances of \(X\). This space was studied in detail by P. S. Urysohn \((^1)\). The problem of P. S. Urysohn that is under consideration here (problem \(\gamma\)) is the following:
Does there exist a continuum whose space of relative distances is zero-dimensional?
It turns out that this problem has a positive solution. Namely:
It is possible to construct a continuum \(C\) such that \(C^*\) is a zero-dimensional space of countable weight.**
Below we give the construction of this continuum.
- Notation. The construction of the continuum \(C\) is carried out in the Hilbert parallelepiped \(H^\omega\), whose points are described by their coordinates:
\[ x=(t_1,t_2,\ldots,t_k,\ldots),\quad 0\leqslant t_k\leqslant \frac1k . \]
Along with \(H^\omega\), the following of its faces are considered:
a) The \(n\)-dimensional parallelepipeds \(H^n\) \((n\geqslant 1)\), formed by the first \(n\) coordinate axes of \(H^\omega\): a point \(x\in H^\omega\) belongs to \(H^n\) if and only if \(t_k=0\) for all \(k>n\). Points of \(H^n\) are described by a set of their \(n\) coordinates:
\[ x=(t_1,t_2,\ldots,t_n). \]
b) The rectangles \(H_n^2\) \((n\geqslant 1)\), formed by the axes \(t_n\) and \(t_{n+1}\): a point \(x\in H^\omega\) belongs to \(H_n^2\) if and only if \(t_k=0\) for all \(k\) different from \(n\) and from \((n+1)\). Points of the rectangle \(H_n^2\) are described by their two coordinates, which, to avoid confusion, are enclosed not in parentheses but in braces:
\[ x=\{t_n,t_{n+1}\}. \]
- Auxiliary construction. Here a continuum \(\Phi_n\) is constructed, lying in the rectangle \(H_n^2\). For its construction, the lower base of this rectangle (i.e. the segment \(0\leqslant t_n\leqslant \frac1n,\ t_{n+1}=0\)) is divided into \(n\) equal parts by the points
\[ x_0,x_1,x_2,\ldots,x_n, \]
where
\[ x_\nu=\left\{\frac{\nu}{n^2},0\right\}. \]
On each segment \([x_{\nu-1},x_\nu]\) \((\nu=1,2,\ldots,n)\)*** such a po—
* The notion of relative distance was introduced by Mazurkiewicz.
** And, consequently, homeomorphic to some subset of the set of irrational numbers.
*** By \([x,x']\) is denoted the segment joining the points \(x\) and \(x'\).
sequence of points
\[ \ldots x_\nu^{-m}, \ldots, x_\nu^{-1}, x_\nu^0, x_\nu^1, \ldots, x_\nu^m, \ldots, \]
such that \(\left(\lim_{m\to-\infty} x_\nu^m\right)=x_{\nu-1}\), \(\left(\lim_{m\to\infty} x_\nu^m\right)=x_\nu\), and, for \(m_2>m_1\), the point \(x_\nu^{m_2}\) lies between the points \(x_\nu^{m_1}\) and \(x_\nu\).
To each \(x_\nu^m\) there corresponds a point \(y_\nu^m\), lying on the upper base \(H_n^2\): if \(x_\nu^m=\{t_n^0,0\}\), then
\[ y_\nu^m=\left\{t_n^0,\frac{1}{n+1}\right\}. \]
By definition,
\[ Z_\nu=\bigcup_{m=-\infty}^{\infty}\left([x_\nu^{2m},y_\nu^{2m+1}]\cup[x_\nu^{2m},y_\nu^{2m-1}]\right); \]
\[ \Phi_n=\left[\bigcup_{\nu=1}^{n} Z_\nu\right]^* . \]
It is easy to show that \(\Phi_n\) is a continuum.
4. The basic construction. Let \(\psi_n\) and \(\varphi_n\) be the natural projections of the parallelepiped \(H^{n+1}\) onto its faces \(H^n\) and \(H_n^2\), respectively: if
\[ x=(t_1,t_2,\ldots,t_n,t_{n+1}) \]
is a point of \(H^{n+1}\), then
\[ \psi_n(x)=(t_1,t_2,\ldots,t_n), \]
\[ \varphi_n(x)=\{t_n,t_{n+1}\}. \]
To an arbitrary compactum \(C_n\) lying in \(H^n\) there is assigned a compactum \(a_n(C_n)\subseteq H^{n+1}\), defined as follows:
\[ a_n(C_n)=\psi_n^{-1}(C_n)\cap\varphi_n^{-1}(\Phi_n). \]
It is easy to show that \(\pi_n=\psi_n|_{a_n(C_n)}\) is a** nearly contractive*** monotone mapping of \(a_n(C_n)\) onto \(C_n\):
\[ \pi_n:a_n(C_n)\to C_n. \]
Let now
\[ C_1=H^1,\quad C_2=a_1(C_1),\ldots,\quad C_{n+1}=a_n(C_n),\ldots \]
The spaces \(C_n\), together with the projections \(\pi_n:C_{n+1}\to C_n\), form a spectrum. By definition,
\[ C=\lim_{\longleftarrow}\{C_n,\pi_n\}. \]
Since all \(C_n\) are compacta, it follows from the connectedness of \(C_1\) and the monotonicity of \(\pi_n\) that \(C\) is a continuum.
5. It can be shown that the continuum \(C\) is the desired one, i.e. that \(C^*\) is a zero-dimensional space of countable weight.
Below are formulated the assertions on which this proof is based.
Definition. Let \(X\) be a topological space and \(x\in X\). A set \(\widetilde U\subseteq X\) containing the point \(x\) is called a relative neighborhood of this point if it is a component of a set open in \(X\).
* Square brackets denote closure.
** If \(f:X\to Y\) and \(A\subseteq X\), then \(f|A\) denotes the same mapping \(f\), but considered only on \(A\).
*** A mapping \(f:X\to Y\), where \(X\) and \(Y\) are metric spaces, is called nearly contractive if the diameter of any \(A\subseteq X\) is not less than the diameter of \(f(A)\).
The system of relative neighborhoods turns \(X\) into a new topological space, denoted below by \(\widetilde X\) and called the space of the relative topology of \(X\).
The interest of this notion is determined by the following theorem:
Theorem 1. If \(X\) is a continuum, then the spaces \(X^*\) and \(\widetilde X\) are homeomorphic.
It can be proved that if the mapping \(\pi : X \to Y\) is continuous, then the mapping induced by it, \(\widetilde\pi : \widetilde X \to \widetilde Y\), is also continuous. Thus, every spectrum \(\{X_n,\pi_n\}\) induces a spectrum \(\{\widetilde X_n,\widetilde\pi_n\}\). The limit space of this spectrum will be denoted below by \(\dot X\) \(\bigl(\dot X=\varprojlim\{\widetilde X_n,\widetilde\pi_n\}\bigr)\).
Theorem 2. If the projections \(\pi_n\) are closed, bicompact and monotone, and \(X=\varprojlim\{X_n,\pi_n\}\) is regular, then \(\widetilde X\) and \(\dot X\) are homeomorphic.
In view of Theorems 1 and 2, it is enough to show that \(\dot C=\varprojlim\{\widetilde C_n,\widetilde\pi_n\}\) is a zero-dimensional space of countable weight. This fact follows from the following theorem:
Theorem 3. Let \(\{X_n,\pi_n\}\) be a spectrum in which all \(X_n\) are metrizable compacta and the \(\pi_n\) are nearly confluent mappings. If:
a) for every \(\varepsilon>0\) there exists an \(n_0\) such that for \(n>n_0\) every point \(x\in X_n\) is contained in a connected set \(F_n\subseteq X_n\) of diameter less than \(\varepsilon\), open and at the same time closed in \(\widetilde X_n\);
b) for every \(n\), \(w(\widetilde X_n)=\aleph_0\),
then \(w(\dot X)=\aleph_0\), \(\operatorname{ind}\dot X=0\)*.
Consideration of the concrete construction of items 3 and 4 makes it possible to establish that the conditions of Theorem 3 are fulfilled in this construction. In checking condition b), the following theorem is used:
Theorem 4. \(w(\widetilde X)>w(X)\) if and only if there exists an open set \(U\subseteq X\) such that the cardinality of the set of components of \(U\) is greater than \(w(X)\).
- One may assert the existence of such a continuum \(F_k\) of arbitrary positive dimension \(k\)***, that \(F_k^*\) is a zero-dimensional space of countable weight. As \(F_k\) it suffices to take the topological product of \(k\) copies of the continuum \(C\) constructed above. The required properties of the space \(F_k^*\) are ensured by the following theorem, in which \(A=\{a\}\) is a set of indices, and the \(X_a\) are arbitrary topological spaces.
Theorem 5. Let \(X=\prod_{a\in A} X_a\), and \(\widehat X=\prod_{a\in A}\widetilde X_a\). If all the \(X_a\), except perhaps finitely many, are connected, then the spaces \(\widehat X\) and \(\widetilde X\) are homeomorphic.
The example of the topological product of countably many simple dyads shows that the connectedness condition in this theorem is essential.
Received
21 I 1963
CITED LITERATURE
- P. S. Uryson, Works on topology and other areas of mathematics, 2, 1951, p. 517.
* \(w(X)\) denotes the weight of the space \(X\).
* \(\operatorname{ind}\) is the small inductive dimension.
** And even infinite-dimensional.