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UDC 513.831
V. V. PROIZVOLOV
A GENERALIZATION OF A THEOREM OF A. N. KOLMOGOROV ON POINTS OF LOCAL TOPOLOGICITY AND ITS CONSEQUENCES
(Presented by Academician P. S. Aleksandrov on 4 IV 1966)
In the paper \((^1)\) the following theorem was proved:
If there is an open countable-to-one mapping \(f\) of a compactum \(X\), then the set \(T_f\) of points of local topologicity of the mapping \(f\) is dense in \(X\)*.
Here we give a generalization of this theorem which makes it possible to assert something new already for spaces with a countable base. In addition, analogues of this generalization are given here for the case of finite-to-one mappings.
Definition. A mapping \(f:X\to Y\) is called preopen if, whenever \(fU\cap fV\ne\Lambda\), where \(U\) and \(V\) are open in \(X\), it follows immediately that \(\langle fU\rangle\cap\langle fV\rangle\ne\Lambda\). Here \(\langle R\rangle\) denotes the set of interior points of \(R\).
Preopen mappings include the class of semiopen** mappings and include the class of open mappings.
Here and below, by a space we mean a Hausdorff space.
Theorem 1. Let there be a locally countable-to-one preopen mapping \(f:X\to Y\), where \(X\) is a locally bicompact space. Then the set \(T_f\) of points of local topologicity of the mapping \(f\) is dense in \(X\).
Proof. Suppose, on the contrary, that there exists an open set \(U\subset X\) with bicompact closure, containing no points of local topologicity, and such that \(f\) is countable-to-one on \(U\). On the basis of this assumption, we shall find a point \(y^\#\in\langle fU\rangle=V\) such that the set \(f^{-1}y^\#\cap U\) is uncountable, which will constitute a contradiction.
Construct a system of open sets and points
\[ \left. \begin{array}{l} x^{(n)}_{i_1\ldots i_n}\in U\\ y^{(n)}\in V\\ U^{(n)}_{i_1\ldots i_n}\subset U\\ V^{(n)}\subset V \end{array} \right\} \quad n=1,2,\ldots;\ i_k=0,1, \]
with the following properties:
\[ \begin{aligned} 1)\quad & f x^{(n)}_{i_1\ldots i_n}=y^{(n)},\\ 2)\quad & x^{(n)}_{i_1\ldots i_n}\in U^{(n)}_{i_1\ldots i_n},\\ 3)\quad & y^{(n)}\in [V^{(n)}],\\ 4)\quad & [V^{(n+1)}]\subset [V^{(n)}],\\ 5)\quad & V^{(n)}\subset \langle fU^{(n)}_{i_1\ldots i_n}\rangle,\\ 6)\quad & [U^{(n+1)}_{i_1\ldots i_n i_{n+1}}]\subset U^{(n)}_{i_1\ldots i_n},\\ 7)\quad & [U^{(n+1)}_{i_1\ldots i_n0}]\cap [U^{(n+1)}_{i_1\ldots i_n1}]=\Lambda. \end{aligned} \]
* Earlier, for the case of finite-to-one mappings, a similar theorem was proved by P. S. Aleksandrov \((^2)\).
** A mapping \(f:X\to Y\) is semiopen if \(\langle fU\rangle\) is nonempty for every open \(U\subset X\).
Suppose that such a system has been constructed. Let us show how everything will follow from this. For any sequence
\[ i_1,i_2,\ldots,i_n,\ldots \tag{1} \]
the set of points \(\{x_{i_1\ldots i_n}^{(n)}\}=A_{i_1\ldots i_n\ldots}\) is such that
\(f x_{i_1\ldots i_n}^{(n)}=y^{(n)}\); \(\{y^{(n)}\}=B\);
\(fA_{i_1\ldots i_n\ldots}=B\). Denote by \(y^\#\) an arbitrary point of complete accumulation of the countable set \(B\), \(y^\#\in [B]\). There exists a point \(x_{i_1\ldots i_n\ldots}\), which is a point of complete accumulation for \(A_{i_1\ldots i_n\ldots}\), such that
\[ f x_{i_1\ldots i_n\ldots}=y^\#;\qquad x_{i_1\ldots i_n\ldots}\in \bigcap_{n=1}^{\infty}[U_{i_1\ldots i_n}^{(n)}]. \]
Such a point \(x_{i_1\ldots i_n\ldots}\) can be found starting from any sequence of the form (1), and from condition 7) it follows that for two different such sequences these points are distinct. Hence \(f^{-1}y^\#\cap U\) is uncountable, and the required contradiction has been obtained.
Let us outline the construction of the system of points and sets possessing properties 1)—7). As \(y^{(1)}\) we take any point of \(V=\langle fU\rangle\) which has at least two points in its inverse image lying in \(U\); denote them by \(x_0,x_1\). As \(U_0^{(1)}\) and \(U_1^{(1)}\) we take neighborhoods of the points \(x_0\) and \(x_1\) such that
\[ [U_0^{(1)}]\cup [U_1^{(1)}]\subset U;\qquad [U_0^{(1)}]\cap [U_1^{(1)}]=\Lambda. \]
As \(V^{(1)}\) we take an arbitrary open set such that
\[ y^{(1)}\in [V^{(1)}],\qquad V^{(1)}\subset V\cap \langle fU_0^{(1)}\rangle \cap \langle fU_1^{(1)}\rangle, \]
which is possible by virtue of the preopenness of \(f\). The further construction is carried out by double induction according to Kolmogorov’s method (with the necessary modifications).
In the theorem, local bicompactness cannot be replaced by completeness in the sense of Čech.* This follows at least from the fact that the space of irrational points of the interval \([0,1]\) can be condensed by a preopen mapping without points of local topologicity onto a certain compactum. However, the following theorem is true:
Theorem \(1'\). Let there be a preopen locally countably multiple mapping \(f:X\to Y\), where \(X\) is either a locally complete space in the sense of Čech or a locally countably compact space. Then the set of points of local condensation of the mapping \(f\) is dense in \(X\).
A point \(x\in X\) is called a point of local condensation of the mapping \(f:X\to Y\) if it has a neighborhood \(Ox\) on which \(f\) is a condensation.
Corollary 1. Let there be a countably multiple preopen mapping \(f:X\to Y\), where \(X\) is either a locally complete space in the sense of Čech or a locally countably compact space. Then the \(\pi\)-weight of \(X\) is equal to the \(\pi\)-weight of \(Y\) (see the definition in (3)).
The corollary is derived from Theorem 1 by means of the following lemma.
Lemma. Let there be a half-open countably multiple mapping \(f:Z\to W\) such that every point \(z\in Z\) is a point of local condensation. Then the \(\pi\)-weight of \(Z\) is equal to the \(\pi\)-weight of \(W\).
If \(\{\omega_\alpha\}\) is a system of open sets dense in \(Z\), then \(\{\langle f\omega_\alpha\rangle\}\) is a dense system of open sets in \(W\), whence it is clear that the \(\pi\)-weight of \(Z\) is \(\ge\) the \(\pi\)-weight of \(W\). Let us prove the reverse inequality. Let \(\omega=\{\omega_\alpha\}\) be a dense system in \(W\); in each \(\omega_\alpha\) fix an arbitrary point \(y_\alpha\in\omega_\alpha\); \(f^{-1}y_\alpha=\{x_{\alpha i}\}\). For every point \(x_{\alpha i}\) take a neighborhood \(Ox_{\alpha i}\) such that \(fOx_{\alpha i}\subseteq\omega_\alpha\), and such that \(f\) is a condensation on \(Ox_{\alpha i}\). All possible pairwise intersections of elements of the system \(\{Ox_{\alpha i}\}\) give us a dense system whose cardinality is equal to the cardinality of the system \(\omega\).
* However, for the case of open mappings this is possible, as B. A. Pasynkov has recently shown.
Corollary 2. Suppose there is a countably multiple preopen mapping \(f: X \to Y\), where \(X\) is a topologically homogeneous bicompactum (or even homogeneous with respect to local weight). Then the weight of \(X\) is equal to the weight of \(Y\).
In connection with the last assertions, let us note that the question of whether the weight of a bicompactum can decrease under a countably multiple open mapping remains unresolved.
Theorem 2. Suppose there is an open finite-to-one mapping \(f: X \to Y\), where \(Y\) is a Čech-complete space. Then in \(X\) there is a dense subset of points of local topologicality of the mapping \(f\).
Lemma 1. A Čech-complete space cannot be represented as the sum of a countable number of nowhere dense subspaces.
The lemma is proved on the basis of the completeness criterion from (4).
Lemma 2. An open \(k\)-to-one mapping is a locally topological mapping.
We outline the proof of the theorem. The space
\[ Y=\bigcup_{i=1}^{\infty}Y_i, \]
where \(Y_i\) is the set of all points of multiplicity \(i\) in \(Y\);
\[ X=\bigcup_{i=1}^{\infty}X_i, \]
where \(X_i=f^{-1}Y_i\). We shall prove that in an arbitrary open set \(U \subset X\) there is a point of local topologicality. Since \(V=fU\) cannot, by Lemma 1, be represented as the sum of a countable number of nowhere dense sets, there exists an \(i\) such that \(Y_i \cap V\) is not nowhere dense in \(V\). For every \(k\), the set
\[ \bigcup_{j=1}^{k}Y_j \]
is closed in \(Y\), and therefore there exists a set \(Q \subset Y_i \cap V\) that is open in \(Y\). Hence, by virtue of the validity of Lemma 2, we conclude that in \(U\) there is a point of local topologicality.
Remark. In the hypothesis of Theorem 2, instead of completeness one may require less of \(Y\). It is sufficient to require that \(Y\) cannot be represented as the sum of a countable number of nowhere dense subspaces, and the theorem will not lose its force. In particular, if \(Y\) is either a locally Čech-complete space or a locally countably compact space, then the theorem remains valid. Moreover, it is sufficient to require from \(f\) local finite multiplicity instead of finite multiplicity.
Theorem \(2'\). Suppose there is a boundedly multiple open mapping \(f: X \to Y\). Then the set \(T_f\) of points of local topologicality of the mapping \(f\) is dense in the space \(X\).
Corollary. If a dyadic space \(X\) is boundedly multiply and openly mapped onto a space with the first axiom of countability, then \(X\) has a countable base.
But there exists a dyadic space that maps finitely and openly onto the rational points of the interval.
Moscow State University
named after M. V. Lomonosov
Received
30 III 1966
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