Mathematics

V. V. PROIZVOLOV

ON DIVIDING MAPPINGS OF LOCALLY CONNECTED SPACES

(Presented by Academician P. S. Aleksandrov on 2 VI 1964)

A mapping \(f\) of a space \(X\) into a space \(Y\) is called dividing if, for any point \(x \in X\) and any neighborhood \(Ox\) of it, there exists a neighborhood \(Oy\) of the point \(y = fx \in Y\) such that its complete preimage is divided into the sum of two disjoint open sets \(G\) and \(H\) in such a way that \(x \in G \subseteq Ox\). This concept, introduced by A. Zarelua \((^1)\), has proved useful in general dimension theory.

All mappings in this paper are continuous, and all spaces are Hausdorff; by a zero-dimensional mapping we mean here a mapping under which the preimage of every point has dimension zero \(\operatorname{ind}\).

Theorem 1. Let there be a dividing mapping \(f : X \to Y\), where \(X\) is a locally connected space and \(Y\) is metrizable. Then the space \(X\) is metrizable.

Proof. The metric space \(Y\), by Bing’s metrization criterion \((^2)\), has a \(\sigma\)-discrete base \(\gamma = \{\gamma_i\}\), where each \(\gamma_i\) is a discrete system of open sets: \(\gamma_i = \{U_{i\alpha}\}\); \(i = 1, 2, \ldots\). A system of open sets in a space is called discrete if every point of the space has a neighborhood meeting only finitely many elements of this system.

Let \(V_{i\alpha}\) denote the \(1/i\)-neighborhood of the set \(U_{i\alpha}\), i.e., the set of all points whose distance from the points of \(U_{i\alpha}\) is no more than \(1/i\). The collection of all \(V_{i\alpha}\) (over all \(i\) and all \(\alpha\)) forms a base of the space \(Y\); denote it by \(\omega = \{V_{i\alpha}\}\).

Observe that every open subset of a locally connected space decomposes into open components. Hence
\[ f^{-1}V_{i\alpha} = \bigcup_{\delta} W_{i\alpha\delta}, \]
where each \(W_{i\alpha\delta}\) is open and connected, and
\[ W_{i\alpha\delta_1} \cap W_{i\alpha\delta_2} = \Lambda,\qquad \delta_1 \ne \delta_2. \]

The union of all \(W_{i\alpha\delta}\) over all indices \(i,\alpha,\delta\) is a system of open subsets of the space \(X\); denote it by \(W\). We shall verify that \(W = \{W_{i\alpha\delta}\}\) is a base in \(X\).

Take an arbitrary point \(x \in X\) with an arbitrary neighborhood \(Ox\) of it. We must show that there is \(W_{i\alpha\delta} \in W\) such that \(x \in W_{i\alpha\delta}\) and \(W_{i\alpha\delta} \subseteq Ox\). Since the mapping \(f\) is dividing, there exists an element \(V_{i\alpha} \in \omega\) such that \(fx \in V_{i\alpha}\) and
\[ f^{-1}V_{i\alpha} = O_1 \cup O_2, \]
where \(O_1\) and \(O_2\) are open in \(X\), with
\[ O_1 \subseteq Ox,\qquad O_1 \cap O_2 = \Lambda. \]
The set \(O_1\) has a component \(W_{i\alpha\delta}\) such that \(W_{i\alpha\delta} \ni x\), and thereby it is proved that \(W\) is a base in \(X\).

Similarly one checks that the system of open sets
\[ Z = \{Z_{i\alpha\delta}\}, \]
where
\[ Z_{i\alpha\delta} = W_{i\alpha\delta} \cap f^{-1}U_{i\alpha}, \]
is also a base in \(X\).

We shall prove that the base \(Z = \{Z_{i\alpha\delta}\}\) is \(\sigma\)-discrete. By Bing’s criterion, this will prove the metrizability of the space \(X\).

Fix \(i\) and consider the system
\[ Z_i = \{Z_{i\alpha\delta}\}; \]
we shall prove that \(Z_i\) is a discrete system of sets in \(X\). First of all, this system is disjoint. Suppose for a moment that it is not discrete, i.e., suppose that there is a point \(x \in X\) such that every neighborhood of it meets infinitely many elements of the system \(Z_i = \{Z_{i\alpha\delta}\}\). Take such a neighborhood \(O_y\) of the point \(y\)

where \(y=fx\), that from the fact that \(Oy\) intersects an element \(U_{i\alpha}\in \gamma_i\) it follows that \(y\in [U_{i\alpha}]\), and there are only finitely many such elements with which \(Oy\) intersects: \(U_{i\alpha_1}, U_{i\alpha_2},\ldots,U_{i\alpha_m}\). And one more requirement for \(Oy\): it is necessary that

\[ Oy \subseteq \bigcap_{k=1}^{m} V_{i\alpha_k}. \]

Now take that component of the set \(f^{-1}Oy\) which contains the point \(x\), and denote it by \(Ox\). For each \(\alpha_k\), \(k=1,2,\ldots,m\), there is a \(\delta_k\) such that \(Ox\subseteq W_{i\alpha_k\delta_k}\). It is not difficult to verify that, in order that \(Ox\cap Z_{i\alpha\delta}\ne \Lambda\), where \(Z_{i\alpha\delta}\in Z_i\), it is necessary that the index \(\alpha\) be equal to one of the \(\alpha_k\), and the index \(\delta\) be equal to one of the \(\delta_k\), \(k=1,2,\ldots,m\). Indeed, if \(Ox\cap Z_{i\alpha\delta}\ne\Lambda\) and \(\alpha\ne\alpha_k\), \(k=1,2,\ldots,m\), then \(Oy\) would not intersect \(U_{i\alpha}\), which would mean that \(Ox\cap Z_{i\alpha\delta}=\Lambda\), since \(Z_{i\alpha\delta}=W_{i\alpha\delta}\cap f^{-1}U_{i\alpha}\), while the neighborhood \(Ox\subseteq f^{-1}Oy\). It is also impossible that \(Ox\cap Z_{i\alpha_k\delta}\ne\Lambda\), and \(\delta\ne\delta_k\), \(k=1,2,\ldots,m\), since from \(Ox\subseteq W_{i\alpha_k\delta_k}\) it follows that \(Ox\cap W_{i\alpha_k\delta}=\Lambda\), if \(\delta\ne\delta_k\), \(k=1,2,\ldots,m\), and hence, all the more, \(Ox\cap Z_{i\alpha_k\delta}=\Lambda\), since \(Z_{i\alpha_k\delta}\subseteq W_{i\alpha_k\delta}\). Consequently, the neighborhood \(Ox\) intersects no more than \(m\) elements of the system \(Z_i\), namely, it is possible that \(Ox\) intersects all or some of the elements \(Z_{i\alpha_k\delta_k}\), where \(k=1,2,\ldots,m\). In view of the arbitrariness of the point \(x\in X\), this means that the system \(Z_i\) is discrete.

Thus, the base \(Z=\{Z_{i\alpha\delta}\}\) is \(\sigma\)-discrete, and the metrizability of the space \(X\) is proved.

A number of important corollaries will be derived from the theorem proved.

Corollary 1. Let there be a zero-dimensional mapping \(f:X\to Y\), where \(X\) is a locally connected and locally bicompact space, and \(Y\) is a metric space. Then the space \(X\) is metrizable.

The mapping \(f\) is a partitioning mapping, since Yu. M. Smirnov noted \((^1)\) that a mapping of a locally bicompact space is partitioning when it is zero-dimensional. After what has been said, Corollary 1 follows directly from the theorem.

Since a closed zero-dimensional mapping of a normal space is partitioning \((^1)\), we have

Corollary 2. Let there be a closed zero-dimensional mapping \(f:X\to Y\), where \(X\) is a locally connected normal space, and \(Y\) is a metric space. Then \(X\) is metrizable.

Corollary 3. Let there be a mapping \(f:X\to Y\), where \(X\) is a locally connected and peripherally bicompact space, and \(Y\) is a metric space. Moreover, let \(f\) be such that \(f^{-1}y\) is a discrete-in-itself set for every point \(y\in Y\). Then \(X\) is metrizable.

Lemma. Let there be a mapping \(f:X\to Y\), where \(X\) is a peripherally bicompact space, and let \(f\) be such that \(f^{-1}y\) is a discrete-in-itself set for every point \(y\in Y\). Then the mapping \(f\) is partitioning.

For an arbitrary point \(x\in X\) and an arbitrary neighborhood \(Ox\) of it, find such a neighborhood \(Oy\) of the point \(y=fx\) that \(f^{-1}Oy\) is split into the sum of two disjoint open sets \(G\) and \(H\) in such a way that \(x\in G\subseteq Ox\). For the point \(x\), take such a neighborhood \(O'x\subseteq Ox\) with a bicompact boundary that \([O'x]\cap f^{-1}y=x\), i.e., in particular, the boundary \(\Gamma\) of the neighborhood \(O'x\) contains no points from \(f^{-1}y\). Denote \(Oy=Y\setminus f\Gamma\); \(G=f^{-1}Oy\cap O'x\), \(H=f^{-1}Oy\setminus G\). It is easy to verify that \(Oy\) is the required neighborhood of the point \(y\).

Taking the lemma into account, Corollary 3 is easily obtained from the theorem.

A base \(\omega\) of the space \(X\) has pointwise power \(\tau\), if \(\tau\) is the minimal cardinal number having the property that, for every point \(x\in X\), the cardinality of the set of elements of the base \(\omega\) containing the point \(x\) is not greater than \(\tau\).

The minimum of the pointwise powers over all bases of the space is called the pointwise weight of the space.

Theorem 2. Let there be a dispersing mapping \(f:X\to Y\), where \(X\) is a locally connected space and \(Y\) is a space of point weight \(\tau\). Then the point weight of the space \(X\) is not greater than \(\tau\).

Proof. Denote by \(\omega=\{\omega_\alpha\}\) a base of the space \(Y\) of point cardinality \(\tau\). Since \(X\) is locally connected, for every \(\alpha\) the set \(f^{-1}\omega_\alpha\) “breaks up” in \(X\) into open components,

\[ f^{-1}\omega_\alpha=\bigcup_\delta V_{\alpha\delta}. \]

The union of all \(V_{\alpha\delta}\) over all indices \(\alpha\) and \(\delta\) forms a base of the space \(X\), as was shown in the proof of Theorem 1; denote this base by \(W=\{V_{\alpha\delta}\}\).

Let us show that the base \(W\) has point cardinality \(\tau\), and this will prove everything. We must verify that an arbitrary point \(x\in X\) belongs to no more than \(\tau\) elements of \(W\). Denote by \(\omega_y=\{\omega_\alpha\}\) the collection of all elements of the base \(\omega\) containing the point \(y=fx\in Y\); the cardinality of \(\omega_y=\{\omega_\alpha\}\), by hypothesis, is not greater than \(\tau\). For every element \(\omega_\alpha\in\omega_y\) there is only one component \(V_{\alpha\delta}\in W\) of the set \(f^{-1}\omega_\alpha\) containing the point \(x\), and consequently the number of all elements \(V_{\alpha\delta}\) of the base \(W\) containing the point \(x\) will be not greater than \(\tau\). The theorem is proved.

Corollary 1. Let there be a zero-dimensional mapping \(f:X\to Y\), where \(X\) is a locally bicompact and locally connected space, and \(Y\) is a space of point weight \(\tau\). Then the point weight of the space \(X\) is not greater than \(\tau\).

Corollary 2. Let there be a closed zero-dimensional mapping \(f:X\to Y\), where \(X\) is a locally connected normal space and \(Y\) is a space of point weight \(\tau\). Then the point weight of the space \(X\) is not greater than \(\tau\).

Corollary 3. Let there be a mapping \(f:X\to Y\), where \(X\) is a locally connected and peripherally bicompact space, and \(Y\) is a space of point weight \(\tau\). Moreover, \(f\) is such that \(f^{-1}y\) is a discrete-in-itself set for every point \(y\in Y\). Then the point weight of the space \(X\) is not greater than \(\tau\).

These corollaries are derived analogously to the way in which the corollaries of Theorem 1 were derived.

Since the point weight of a bicompactum coincides with its integral weight \((^3)\), Corollary 1 immediately implies the well-known theorem of Mardešić:

A zero-dimensional mapping of a locally connected bicompactum cannot lower its weight.

Moscow State University
named after M. V. Lomonosov

Received
29 V 1964

CITED LITERATURE

\(^{1}\) A. Z. Melua, DAN, 144, No. 4, 713 (1962).
\(^{2}\) R. H. Bing, Canad. J. Math., 3, No. 2, 175 (1951).
\(^{3}\) A. Mishchenko, DAN, 144, No. 5, 985 (1952).