I. K. LIFANOV

ON TWO PROBLEMS OF MARDEŠIĆ

(Presented by Academician P. S. Aleksandrov on 23 XII 1964)

MATHEMATICS

1. In this paper two examples are constructed.

Example 1. A locally connected continuum* \(X^*\), connected by ordered continua, of dimension \(\operatorname{Ind} X^* = 1\), which is not a continuous image of any ordered continuum.

Example 2. A locally connected continuum \(X^{**}\), connected by ordered continua, for which \(\dim X^{**} = 1\), while \(\operatorname{ind} X^{**} = \operatorname{Ind} X^{**} = 2\).

Remark 1. These examples give an answer to two questions of Mardešić from paper \((^2)\) (problems 15 and 17).

Remark 2. In the construction of Example 2 one remarkable idea from a paper of O. V. Lokutsievskii \((^3)\) is used.

1. We proceed to the construction of the examples.

A. Canonical decomposition of a Cantor perfect set \(C\). If a point \(x\) of the Cantor perfect set \(C\) is an endpoint of a complementary interval, then we identify this point \(x\) with the other endpoint of the same complementary interval. Every other point \(x\) of the Cantor perfect set is, by definition, a one-point element of the decomposition. The indicated decomposition of the Cantor perfect set is called the canonical one of zero rank. Obviously, this decomposition is continuous, and its space \(Z_0\) is homeomorphic to an interval.

By a complementary interval of rank \(n\), \(n = 1, 2, \ldots\), to the Cantor set \(C\) we shall mean any complementary interval of length \(1/3^n\).

* A connected bicompactum is called a continuum. An ordered bicompactum (in particular, a continuum) is an ordered set which in its order topology is a bicompactum (continuum). A bicompactum \(X\) is called connected by ordered continua if for any two of its points \(x_1\) and \(x_2\) there exists an ordered continuum \(X_0\), topologically contained in \(X\), such that \(x_1\) is its first point and \(x_2\) is the last point in the order topology of this continuum \(X_0\). Obviously, connectedness by ordered continua implies connectedness. If a topological space is represented as a sum of pairwise nonintersecting (disjoint) closed sets \(A_\alpha\), then one says that a decomposition \(\{A_\alpha\}\) of the space \(X\) is given. A decomposition \(\{A_\alpha\}\) of a topological space \(X\) is called continuous (see \((^4)\)) if, for every element \(A_0\) of the decomposition \(\{A_\alpha\}\) and every neighborhood \(U A_0\) of it, there is a marked neighborhood \(U'A_0 \subseteq U A_0\) of this element \(A_0\); here a set \(M \subset X\) is called marked for the decomposition \(\{A_\alpha\}\) if it is a sum of some elements of this decomposition. If a decomposition \(\{A_\alpha\}\) of a space \(X\) is given, then the mapping \(f\) which assigns to each point \(x \in X\) the element of the decomposition containing it is called the natural mapping of the space \(X\) onto the set of all elements \(A_\alpha\) of the decomposition \(\{A_\alpha\}\). A topology is introduced on this set: a set \(\mathfrak{A}\) of elements of the decomposition \(\{A_\alpha\}\) is considered open if the set \(\bigcup_{A_\alpha \in \mathfrak{A}} A_\alpha\) is open in the space \(X\). It is known (see \((^4)\)) that the space \(Z\) of a continuous decomposition \(\{A_\alpha\}\) of a bicompactum \(X\) will also be a bicompactum.

The P. S. Aleksandrov line, or transfinite line: to each ordinal number \(\alpha < \omega_1\) we put in correspondence a copy \(I_\alpha\) of the interval \([0,1]\), identifying its left endpoint with \(\alpha\) and its right endpoint with \(\alpha + 1\). The set obtained by adjoining to the indicated number \(\omega_1\) the generalized segment \(P\) is topologized by establishing the order relation. For distinct \(\xi_1, \xi_2 \in P \setminus (\omega_1)\) we put \(\xi_1 < \xi_2\) if one of the following conditions is fulfilled: 1) \(\xi_1 \in I_{\alpha_1}\), \(\xi_2 \in I_{\alpha_2}\), where \(\alpha_1 < \alpha_2\); 2) \(\xi_1, \xi_2 \in I_\alpha\), and on this interval \(\xi_1 < \xi_2\). In addition, by definition, \(\omega_1 > \xi\) for every \(\xi \in P \setminus (\omega_1)\). It is easy to see that \(P\) is a continuum. This continuum \(P\) is also called the P. S. Aleksandrov line, or the transfinite line.

If in the Cantor set \(C\) we identify the endpoints of adjacent intervals of rank \(> n\), \(n=1,2,\ldots\), regarding all other points of \(C\) as one-point elements of the partition, then we obtain the canonical partition of rank \(n\). The canonical partition of any rank \(n\) is continuous, and its space \(Z_n\) is homeomorphic to the discrete sum of \(2^n\) intervals.

B. Construction of the bicompactum \(X^*\), example 1. Consider the bicompactum
\(R=P\times C\). The desired bicompactum \(X^*\) is obtained from \(R\) as the space of a certain continuous partition \(D\). We define the partition \(D\) on each set of the form \(C_\xi=(\xi\times C)\subseteq R\), where \(\xi\) is a point of the bicompactum \(P\). Let \(\xi\) be a transfinite number. Then on the set \(C_\xi\) we define our partition as the canonical partition of zero rank of the Cantor set \(C_\xi\equiv C\). Let \(\xi\) be a dyadic-rational number of an interval \(I_\alpha\), say \(\xi=m/2^k\) (\(m\) odd). Then on the set \(C_\xi\) we define the partition \(D\) as the canonical partition of rank \(k-1\). If \(\xi\) is a dyadic-irrational number of some interval \(I_\alpha\), then on the set \(C_\xi\) every point \((\xi,x)\in C_\xi\) is a (one-point) element of the partition \(D\). The partition space \(D\) is, by definition, our bicompactum \(X^*\). We prove successively the properties of the space \(X^*\).

\(1^\circ.\) The partition \(D\) of the bicompactum \(R\) is continuous, and the space of this partition is a locally connected bicompactum.

Proof. Let \(A_0\) be an arbitrary element of the partition \(D\), and let \(UA_0\) be an arbitrary neighborhood of it in the bicompactum \(R\). We take a neighborhood \(U_1A_0\) of the same element of the partition, contained in \(UA_0\), of the form
\(U_1A_0=U_1\times V_1\), where \(U_1\) is an interval of the ordered space \(P\), and \(V_1\) is open in \(C\). Denote by \(\pi_P\) the projection of the bicompactum \(R\) onto the bicompactum \(P\), and by \(\pi_C\) the projection of the bicompactum \(R\) onto \(C\). Further, denote by \(f_k\) the natural mapping of the set \(C\) onto the space \(Z_k\) of its canonical partition of rank \(k\). By \(f_0\) we denote the natural mapping of \(C\) onto the space \(Z_0\) of its canonical partition of zero rank. Transfinite numbers \(\xi\in P\) and the points \(1/2\) of the intervals \(I_\alpha\) we shall call points of zero rank, and dyadic-rational points \(\xi=m/2^k\) (\(m\) not divisible by 2) of the intervals \(I_\alpha\) we shall call points of rank \(k-1\). Further, all dyadic-irrational points \(\xi\) of the intervals \(I_\alpha\) we shall call points of infinite rank. Now take, in the neighborhood \(U_1\) of the point \(\pi_PA_0\) in \(P\), a point \(\xi=m/2^k\) of least rank \(k-1\). Further, take, in the neighborhood \(V_1\) of the element \(\pi_CA_0\) of the canonical partition of rank \(k-1\), a neighborhood \(V_2\subseteq V_1\) of the same element \(\pi_CA_0\), marked for this canonical partition, in such a way that the set \(f_{k-1}V_2\) is connected in the space \(Z_{k-1}\). Such a neighborhood \(V_2\) of the element \(\pi_CA_0\) can be found, since the canonical partition of rank \(k-1\) is continuous, and the space of this partition is locally connected. Consider the neighborhood \(U_2A_0=U_1\times V_2\) of the element \(A_0\) of the partition \(D\) of the bicompactum \(R\); it is easily checked that it is marked for the partition \(D\), whence it follows that the partition \(D\) is continuous.

The space \(X^*\) of the partition \(D\) is a bicompactum; we prove its local connectedness. For this it is enough to show that the image, under the natural mapping \(f:R\to X^*\), of the neighborhood \(U_2A_0\) constructed above will be a connected neighborhood \(fU_2A_0\) of the point \(A_0\) in the bicompactum \(X^*\).

Consider the set \(C_\xi=(\xi\times C)\), \(\xi=m/2^k\). We shall first show that \(f(C_\xi\cap U_2A_0)\) is connected. For this, in turn, it is enough to prove that the set \(f(C_\xi\cap U_2A_0)\) is homeomorphic to the set \(f_{k-1}(C_\xi\cap U_2A_0)\). But this follows from the fact that on the set \(C_\xi\equiv(\xi\times C)\) our partition of the bicompactum \(R\) is precisely the canonical partition of rank \(k-1\) of the set \(C_\xi\equiv C\), and the neighborhood \(U_2A_0=U_1\times V_2\) has been chosen so that \(V_2\) is marked for this partition, and the set \(f_{k-1}V_2\) is connected in \(Z_{k-1}\subseteq X^*\). Consequently, the set
\(f(C_\xi\cap U_2A_0)=f_{k-1}(C_\xi\cap U_2A_0)\)
is connected. We now prove the connectedness of the set \(fU_2A_0\), which is open in \(X^*\). Suppose that there is a subset \(V\subset fU_2A_0\), open-and-closed in \(fU_2A_0\). Since \(f(C_\xi\cap U_2A_0)\) is connected, it is necessary...

\(f(C_{\xi}\cap U_2A_0)\subseteq V\). Then \(f^{-1}V\) is a (proper) clopen subset of the set \(U_2A_0\), and \(C_{\xi}\cap U_2A_0\subset f^{-1}V\). Then the set \(\pi_P(U_2A_0\setminus f^{-1}V)\) is a proper subset of the connected neighborhood \(U_1\) in \(P\) and is clopen in \(U_1\), which contradicts the connectedness of \(U_1\). Assertion \(1^0\) is proved.

\(2^0.\ \dim X^*=\operatorname{ind}X^*=1.\)

It remains to prove the inequality \(\operatorname{ind}X^*\le 1\). For this it is only necessary to choose the neighborhood \(U_2A_0\) of the element \(A_0\) of the decomposition \(D\), constructed in \(1^0\), in such a way that in the neighborhood \(U_1\) the ends are dyadic irrational, i.e. points of infinite rank, while the two endpoints of the neighborhood \(V_2\) in \(C\) are ends of some adjacent intervals of the set \(C\) (generally speaking, of different ranks). Then \(f\,\operatorname{fr} U_2A_0=\operatorname{fr} fU_2A_0\) and \(\operatorname{ind}\operatorname{fr} fU_2A_0\le 0\).

\(3^0.\) The bicompactum \(X^0\) is connected by ordered continua. Let \(A_1\) and \(A_2\) be two distinct points of the bicompactum \(X^*\). Consider the sets \(A_1\subseteq R\), \(A_2\subseteq R\), and the points \(\xi_1=\pi_PA_1\), \(\xi_2=\pi_PA_2\) of the continuum \(P\). Let \(\xi_2>\xi_1\). Further, let \(\xi_0\) be the first transfinite number following \(\xi_2\). Consider the ordered continuum \(F_1=f((\xi_1,x_1)\cup (U_1\times x_1))\), where \(x_1\) is any one point of the set \(\pi_C\xi_1\), and \(U_1\) is the half-interval \((\xi_1,\xi_0]\) of the ordered continuum \(P\). The first point of this ordered continuum will be the point \(A_1\), and the last will be the point \(f(\xi_0,x_1)\). Next, on the segment \(fC_{\xi_0}\) consider the smaller segment \(F_2\), whose first point will be the point \(f(\xi_0,x_1)\), and whose last will be the point \(f(\xi_0,x_2)\), where \(x_2\) is any one (fixed) point of the set \(\pi_C\xi_2\). Finally, consider the ordered continuum \(F_3=f((\xi_0,x_2)\cup (U_2\times x_2))\), where \(x_2\) is the same fixed point of the set \(\pi_C\xi_2\), and \(U_2\) is the half-interval \([\xi_2,\xi_0)\) of the ordered continuum \(P\). The first point of \(F_3\) will be \(f(\xi_0,x_2)\), and the last \(A_2=f(\xi_2,x_2)\). Then \(F_0=F_1\cup F_2\cup F_3\) is an ordered continuum whose first point is \(A_1\), and whose last is \(A_2\).

We pass to the proof of the main assertion.

\(4^0.\) The bicompactum \(X^*\) is not a continuous image of any ordered continuum.

Proof. Suppose the contrary: let there be an ordered continuum \(T\) and a continuous mapping \(g:T\to X^*\). Consider the auxiliary bicompactum \(L=P\times[0,1]\). We shall prove that the bicompactum \(X^*\) is mapped continuously onto the bicompactum \(L\). Indeed, the bicompactum \(L\) is homeomorphic to the space of the following decomposition of the bicompactum \(R\): on each set \(C_{\xi}=\xi\times C\), where \(\xi\) is an arbitrary point of \(P\), we define the auxiliary decomposition \(\Delta\) of the bicompactum \(R\) as the canonical decomposition of null rank. Now it is obvious that there exists a continuous mapping \(h\) of the bicompactum \(X^*\) onto the space \(Z=L\) of the decomposition \(\Delta\). The continuous mapping \(hg\) maps the ordered continuum \(T\) onto the bicompactum \(L=P\times[0,1]\). The following theorem is known (see \((^1)\)): if there exists a continuous mapping of an ordered continuum \(T\) onto the product of two locally connected continua (in our case \(L\) is the product of the locally connected continua \(P\) and \([0,1]\)), then each of the continua (in our case the continua \(P\) and \([0,1]\)) must have Suslin’s property.* But the continuum \(P\) does not have Suslin’s property. The contradiction obtained proves assertion \(4^0\).

1. The bicompactum \(X^{**}\) of Example 2 will be constructed by literally repeating the arguments and constructions of work \((^3)\).

In the bicompactum \(X^*\) consider the set \(F=fC_{\omega_1}\), homeomorphic to the interval \([0,1]\). The set of one-sided points of the Cantor set of the perfect set \(\omega_1\times C\) will be denoted by \(C_{\omega_1}^{0}\), and \(fC_{\omega_1}^{0}\) by \(M\). The set \(M\), evidently, is everywhere dense in \(F\). Consider \(Q=X_1^*\cup X_2^*\), where \(X_1^*, X_2^*\) are nonintersecting copies of the space \(X^*\). By \(F_i, M_i\) we shall denote—

* A bicompactum \(X\) has Suslin’s property if every system of pairwise disjoint open sets in it is at most countable.

denote the sets lying in \(X_i^*,\ i=1,2,\) and corresponding to the prescribed sets \(F, M\) in \(X^*\). Choose on \(F_2\) an arbitrary everywhere dense set \(N\) of the same order type as \(M_1\) and having no points in common with \(M_2\), except for the endpoints of \(F_2\). As is known (see \((^5)\)), there exists a similarity mapping \(g\) of the segment \(F_2\) onto \(F_1\) which carries \(N\) into \(M_1\). One may regard \(g\) as a continuous mapping of \(Q\) onto a certain bicompactum \(X^{**}=gQ\). Obviously, \(g(X_1^*)=S_1,\ g(X_2^*)=S_2\) are homeomorphisms, \(X^{**}=S_1\cup S_2\). If \(E=gF_1=gF_2\), then \(S_1\cap S_2=E\). The bicompactum \(X^{**}\) is the desired one, i.e., it is locally connected, connected by ordered continua, has dimension \(\dim X^{**}=1\), and \(\operatorname{ind} X^{**}=\operatorname{Ind} X^{**}=2\). Moreover, \(\operatorname{ind} X^{**}\ne \max\{\operatorname{ind} S_1,\operatorname{ind} S_2\}\). The fact that the bicompactum is locally connected and connected by ordered continua is proved in exactly the same way as the analogous assertions for the bicompactum \(X^*\) of Example 1. (For the proof that \(\dim X^{**}=1,\ \operatorname{ind} X^{**}=\operatorname{Ind} X^{**}=2\), see \((^3)\).)

In conclusion I express my gratitude to my adviser V. I. Ponomarev for his assistance in this work.

Moscow State University
named after M. V. Lomonosov

Received
22 X 1964

CITED LITERATURE

\(^1\) S. Mardešić, Glasnik mat.-fiz. i astron., 15, 85 (1960). \(^2\) S. Mardešić, P. Papić, Matematička biblioteka, 25 (1963). \(^3\) O. V. Lokutsievskii, DAN, 67, No. 2, 217 (1949). \(^4\) P. Alexandroff, H. Hopf, Topologie, Berlin, 1935. \(^5\) B. Hausdorff, Set Theory, 1937.