MATHEMATICS

O. V. LOKUTSIEVSKII

ON THE TOPOLOGY OF CONTINUA

(Presented by Academician P. S. Aleksandrov, 23 III 1965)

The relative distance \(\widetilde{\rho}_X(x_1,x_2)\) between points \(x_1\) and \(x_2\) of a continuum \(X\), introduced by C. Mazurkiewicz \((^4)\), is the lower bound of the diameters of those of its connected subsets which contain both these points. The relative distance turns \(X\) into a new metric space—the space of the relative metric of the continuum \(X\). This space (denoted below by \(\widetilde{X}\)) was investigated by P. S. Urysohn \((^5)\). To him also belongs the formulation of the following three problems:

Problem \(\gamma\). Does there exist a continuum \(X\) such that the space \(\widetilde{X}\) is zero-dimensional?

Problem \(\delta\). Does there exist a continuum \(X\) such that the space \(\widetilde{X}\), while not being zero-dimensional, contains no connected subsets other than one-point sets?

Problem \(\varepsilon\). Does there exist a continuum \(X\) such that the space \(\widetilde{X}\) is cohesive, while not being connected?

The first of these problems received a positive solution in the author’s paper \((^2)\), and the third—in a paper of M. A. Shtan’ko \((^6)\).

Below a one-dimensional continuum \(\Delta\) is constructed, realizing a positive solution of problem \(\delta\).

1. Let us call a continuum \(X\) degenerate if \(w\widetilde{X}=\aleph_0\)*, \(\dim \widetilde{X}=0\). The basis of the induction process leading to the continuum \(\Delta\) is the existence, established in \((^2)\), of degenerate continua. The degenerate continuum \(C\) constructed in \((^2)\) is snakelike and, consequently, homeomorphic to a part of the plane (see \((^1)\)). Therefore:

In any triangle \([x_1,x_2,x_3]\) there can be found a degenerate continuum containing its vertices \(x_1\) and \(x_2\).

1. Consider the regular tetrahedron

\[ \Delta_0=[a,b,c,p], \]

the length of each of whose edges is equal to unity. At the first step of the induction there will be constructed a continuum \(\Delta_1\) lying in this tetrahedron. For its construction, choose first of all on the segment \([a,p]\) a sequence of points \(a_1,a_2,\ldots,a_m,\ldots\), determined by the equalities \(\rho(a_m,a)=2^{-m}\). In an analogous way the points \(b_m\) and \(c_m\) \((m=1,2,\ldots)\), lying respectively on the segments \([b,p]\) and \([c,p]\), are chosen.

The continuum \(\Delta_1\) will be defined as the sum of two sets: a degenerate continuum \(\Gamma\) and a compactum \(\Sigma_0\).

For the construction of the first of these, consider the triangle \(\tau_c=[a,b,p]\). The point of intersection of the segment \([a_{m+1},b_m]\) with the perpendicular dropped from the point \(a_m\) to the segment \([a_{m+1},b_{m+1}]\) will be denoted by \(q_m\). According to point 1, there exists a degenerate continuum \(A_m\), lying in

* By \(w\widetilde{X}\) is denoted the weight of the space \(\widetilde{X}\).

triangle \([a_m,a_{m+1},q_m]\) and containing the points \(a_m\) and \(a_{m+1}\). We set

\[ A=(a)\cup\left(\bigcup_{m=1}^{\infty} A_m\right). \]

By \(B\) we denote the continuum symmetric to the continuum \(A\) with respect to the line joining the point \(p\) to the midpoint of the segment \([a,b]\). It can be shown that \(A\) and \(B\) are expressible continua and that

\[ \lim_{m\to\infty} \widetilde{\rho}_A(a_m,a)=0,\qquad \lim_{m\to\infty} \widetilde{\rho}_B(b_m,b)=0. \tag{1} \]

Let \(\Phi\) be any expressible continuum lying in the triangle \(\tau_p=[a,b,c]\) and containing its vertices \(a\) and \(b\). By definition,

\[ \Gamma=A\cup \Phi\cup B. \]

It is easy to show that \(\Gamma\) is an expressible continuum.

We proceed to the construction of the compactum \(\Sigma_0\). To this end, having fixed a positive integer \(m\), we project the continuum \(\Phi\) from the point \(p\) onto the triangle \([a_m,b_m,c_m]\). The resulting continuum \(\Phi_m\), obviously, contains the points \(a_m\) and \(b_m\). And since the described projection is a \(2^{-m+1}\)-shift, we have

\[ \Phi_m \subset O(\Phi,2^{-m+1}). \tag{2} \]

There exists a finite system \(T_m^1,T_m^2,\ldots,T_m^{k(m)}\) of regular triangles lying in the triangle \([a_m,b_m,c_m]\), \(T_m^k=[a_m^k,b_m^k,c_m^k]\), of diameter less than \(2^{-(m+3)}\), such that:

\[ (\alpha_1)\quad a_m^1=a_m,\quad b_m^{k(m)}=b_m. \]

\[ (\alpha_2)\quad T_m^k\cap T_m^{k'}= \begin{cases} a_m^{k'}=b_m^k, & \text{if } k'=k+1,\\ \Lambda, & \text{if } |k'-k|>1. \end{cases} \]

\[ (\alpha_3)\quad \text{The set } Z_m=\bigcup_{k=1}^{k(m)} T_m^k \text{ lies in } O(\Phi_m,2^{-(m+3)}). \]

Each triangle \(T_m^k=[a_m^k,b_m^k,c_m^k]\) is the base of two regular tetrahedra lying in \(\Delta_0\). Let \((\Delta_0)_m^k\) be the one of them whose fourth vertex (denote it by \(p_m^k\)) is separated from the point \(p\) by the plane of the triangle \([a_m,b_m,c_m]\):

\[ (\Delta_0)_m^k=[a_m^k,b_m^k,c_m^k,p_m^k]. \]

From \((\alpha_2)\) it follows that the set

\[ (\Sigma_0)_m=\bigcup_{k=1}^{k(m)}(\Delta_0)_m^k \]

is a continuum, and moreover, in accordance with (2) and \((\alpha_3)\), the inclusion
\((\Sigma_0)_m\subset O(\Phi,2^{-m+2})\) holds. Thus,

\[ \Sigma_0=\Phi\cup\left(\bigcup_{m=1}^{\infty}(\Sigma_0)_m\right) \]

turns out to be a compactum. The components of this compactum are, first, all the sets \((\Sigma_0)_m\), and second, the set \(\Phi\). We set

\[ \Delta_1=\Gamma\cup\Sigma_0. \]

It follows from \((\alpha_1)\) that \(\Delta_1\) is a continuum. The first step in the construction of the continuum \(\Delta\) is complete.

1. To continue the induction process it is convenient to use the system of mappings

\[ f_m^k:\Delta_0\to(\Delta_0)_m^k\quad (k=1,2,\ldots,k(m);\ m=1,2,\ldots), \]

each of which is a linear homeomorphism taking the vertices \(a,b,c,p\) of the tetrahedron \(\Delta_0\), respectively, to the vertices \(a_m^k,b_m^k,c_m^k,p_m^k\) of the tetrahedron \((\Delta_0)_m^k\).

Let \(E\) be an arbitrary subset of the tetrahedron \(\Delta_0\). We put

\[ \varphi E=\Gamma\cup\left(\bigcup_{m=1}^{\infty}\bigcup_{k=1}^{k(m)} f_m^k E\right) \]

(thus, in particular, \(\Delta_1=\varphi\Delta_0\)). If \(E\) is a continuum containing the points \(a\) and \(b\), then \(\varphi E\) is also a continuum containing the points \(a\) and \(b\). Moreover, it is clear that from the inclusion \(E'\subseteq E\) there follows the inclusion \(\varphi E'\subseteq \varphi E\). Therefore, putting

\[ \Delta_{n+1}=\varphi\Delta_n \qquad (n=0,1,2,\ldots), \]

we obtain a decreasing sequence of continua, each of which contains the points \(a\) and \(b\). By definition,

\[ \Delta=\bigcap_{n=0}^{\infty}\Delta_n. \]

It can be shown that \(\varphi\Delta=\Delta\), and, consequently, \(\Delta\) is representable in the form of the sum

\[ \Delta=\Gamma\cup\left(\bigcup_{m=1}^{\infty}\bigcup_{k=1}^{k(m)} \Delta_m^k\right), \]

in which \(\Delta_m^k=f_m^k\Delta\) are continua homeomorphic to \(\Delta\).

We describe a plan by following which one can establish that the continuum \(\Delta\) indeed realizes a positive solution of problem \(\delta\).

1. Let \(G\) and \(H\) be open subsets of the space \(\widetilde{\Delta}\) such that \(a\in G\), \(b\in H\), and \(G\cup H=\widetilde{\Delta}\). Using relations (1), one can establish the existence of such indices \(m\) and \(k\) \((k\leq k(m))\) that \(f_m^k a\in G\), \(f_m^k b\in H\). Hence, by induction, it follows that the points \(a\) and \(b\) cannot be separated in the space \(\widetilde{\Delta}\) by the empty set, and, thus,

The small inductive dimension of the space \(\widetilde{\Delta}\) is positive.

1. Definition (see (3)). A continuum \(Y\) is called correctly situated in the continuum \(X\) containing it if the identity mapping \(\xi:Y\to X\) is a homeomorphism.

It turns out that both \(\Gamma\) and all the \(\Delta_m^k\) are correctly situated in \(\Delta\). In other words, the following (topologically understood) equality holds:

\[ \widetilde{\Delta}=\widetilde{\Gamma}\cup\left(\bigcup_{m=1}^{\infty}\bigcup_{k=1}^{k(m)} \widetilde{\Delta}_m^k\right). \tag{3} \]

Notation. The component of a metric space \(X\) containing a point \(x\in X\) will be denoted by \(\varkappa_x X\).

Since \(\operatorname{ind}\widetilde{\Gamma}=0\), relation (3) makes it possible to establish that \(\varkappa_a\widetilde{\Delta}=a\), \(\varkappa_b\widetilde{\Delta}=b\). These equalities, together with the same relation (3), lead, in turn, to the following assertion:

Let \(x\) be an arbitrary point of \(\widetilde{\Delta}\). If there exist indices \(m\) and \(k\) \((k\leq k(m))\) such that \(x\in\widetilde{\Delta}_m^k\), then \(\varkappa_x\widetilde{\Delta}=\varkappa_x\widetilde{\Delta}_m^k\); otherwise \(\varkappa_x\widetilde{\Delta}=x\).

It follows from this assertion that:

The space \(\hat{\Delta}\) contains no connected subsets except one-point ones.

1. Let

\[ \Gamma^{k_1k_2\ldots k_n}_{m_1m_2\ldots m_n} = \left(f^{k_1}_{m_1}\cdot f^{k_2}_{m_2}\cdots f^{k_n}_{m_n}\right)\Gamma, \]

\[ \Delta^{k_1k_2\ldots k_n}_{m_1m_2\ldots m_n} = \left(f^{k_1}_{m_1}\cdot f^{k_2}_{m_2}\cdots f^{k_n}_{m_n}\right)\Delta . \]

Denote by \(P\) the sum of all sets of the form \(\Gamma^{k_1k_2\ldots k_n}_{m_1m_2\ldots m_n}\) (including \(\Gamma\)), and put \(Q=\Delta\setminus P\). It is easy to see that a point \(x^0\) of \(\Delta\) belongs to \(Q\) if and only if, for every integer \(n\), there exists a set containing it of the form \(\Delta^{k_1k_2\ldots k_n}_{m_1m_2\ldots m_n}\). What has been said allows us to establish that:

(a) The continuum \(\Delta\) is locally connected at each of its points belonging to \(Q\).

(b) The set \(Q\), considered in the metric of the continuum \(\Delta\), is zero-dimensional.

From (a) it follows that

(c) The identity mapping \(\xi:\tilde{\Delta}\to\Delta\) is a homeomorphism on the set \(Q\).

Let the sets \(P\) and \(Q\), considered in the metric of the space \(\tilde{\Delta}\), be denoted respectively by \(P_{\tilde{\Delta}}\) and \(Q_{\tilde{\Delta}}\). From (b) and (c) it follows that

(d) \(w(Q_{\tilde{\Delta}})=\aleph_0;\quad \operatorname{ind}(Q_{\tilde{\Delta}})=0.\)

Now observe that the weight of each of the spaces \(\tilde{\Gamma}^{k_1k_2\ldots k_n}_{m_1m_2\ldots m_n}\) is countable and that they are all closed in \(\tilde{\Delta}\).* Therefore

(e) \(w(P_{\tilde{\Delta}})=\aleph_0;\quad \operatorname{ind}(P_{\tilde{\Delta}})=0.\)

From (d) and (e) we conclude that \(w\tilde{\Delta}=\aleph_0\) and that \(\operatorname{ind}\tilde{\Delta}\leq 1\). And since, according to item 4, \(\operatorname{ind}\tilde{\Delta}>0\), it follows that

The space \(\tilde{\Delta}\) is one-dimensional and has countable weight.

Received
19 III 1965

CITED LITERATURE

\({}^{1}\) R. H. Bing, Duke Math. J., 18, 653 (1951).
\({}^{2}\) O. V. Lokutsievskii, DAN, 151, No. 4 (1963).
\({}^{3}\) O. V. Lokutsievskii, DAN, 157, No. 5 (1964).
\({}^{4}\) S. Mazurkiewicz, Fund. Math., 1 (1920).
\({}^{5}\) P. S. Urysohn, Works on Topology and Other Fields of Mathematics, 2, 1951, p. 517.
\({}^{6}\) M. A. Shtan’ko, UMN, 18, issue 5 (113) (1963).

* The latter follows from the compactness of \(\Gamma^{k_1k_2\ldots k_n}_{m_1m_2\ldots m_n}\) and from the continuity of the identity mapping \(\xi:\tilde{\Delta}\to\Delta\).