UDC 513.831

MATHEMATICS

V. V. PROIZVOLOV

ON FINITELY MULTIPLE OPEN MAPPINGS

(Presented by Academician P. S. Aleksandrov, May 7, 1965)

Here finitely multiple and \(k\)-multiple mappings of topological spaces are considered. It has turned out that finitely multiple open mappings constitute an important case of zero-dimensional mappings, naturally singled out into a special class, since assertions valid for finitely multiple mappings are false, for example, for compact** mappings.

The main result should be considered Theorem 1, which apparently does not hold for compact open mappings; this, however, has remained unknown. It is useful to compare Theorem 1 with S. Mardešić’s theorem stating that the weight of a locally connected bicompactum is not decreased under a zero-dimensional mapping, and also with its strengthenings given in \((^6)\).

Theorem 1. Under a finitely multiple open mapping the weight of a locally bicompact space is not decreased.*** *

Lemma. Let there be an open \(k\)-multiple mapping \(f\) of a space \(X\) onto \(Y\). Then the weight of \(Y\) is not less than the weight of \(X\).

We prove the lemma. For every point \(y \in Y\) there exists a neighborhood \(O\) such that

\[ f^{-1}Oy=\bigcup_{i=1}^{k} Ox_i,\qquad Ox_{i_1}\cap Ox_{i_2}=\Lambda,\quad i_1\ne i_2, \]

where \(\bigcup_{i=1}^{k} x_i=f^{-1}y\), and such that \(fOx_i=Oy,\ i=1,\ldots,k\). Indeed, if \(O'x_i,\ i=1,\ldots,k,\) are pairwise disjoint neighborhoods, then one may put

\[ Oy=\bigcap_{i=1}^{k} fO'x_i, \]

and then \(Ox_i=O'x_i\cap f^{-1}Oy,\ i=1,\ldots,k\). Note that on each \(Ox_i\) the mapping \(f\) is topological.

Such a neighborhood \(Oy\) will be called a marked neighborhood of the point \(y\). The system \(u_y=\{O_\alpha y\}\) of all marked neighborhoods of the point \(y\) forms a base at this point, as is not hard to verify. The system \(u=\{u_y\}\), over all \(y\in Y\), forms a base of the space \(Y\).

From the base \(u\) choose a base \(v\), whose cardinality is equal to the weight of \(Y\), \(v=\{v_\alpha\}\), where every element \(v_\alpha\in v\) is a marked neighborhood of some point \(y\in Y\). As we have seen,

\[ f^{-1}v_\alpha=\bigcup_{i=1}^{k} w_{\alpha i}, \]

where each \(w_{\alpha i}\) is open in \(X\), and by means of \(f\) it is mapped topologically onto all of \(v_\alpha\).

The cardinality of the system \(w=\{w_{\alpha i}\}\) over all admissible \(\alpha\) and \(i\) is obviously equal to the weight of \(Y\). We shall show that all possible pairwise intersections of elements of \(w\) form a base of the space \(X\), which will complete the proof of the lemma.

Let \(Ox\) be an arbitrary neighborhood of some point \(x\in X\). There exists \(v_\alpha\in v\) such that \(y\in v_\alpha\subseteq fOx\), where \(y=fx\). Choose \(w_{\alpha i}\) such that \(x\in w_{\alpha i}\). Denote

\[ Oy=f(w_{\alpha i}\cap Ox); \]

there exists \(v_\beta\in v\) such that \(x\in v_\beta\subseteq Oy\). Choose \(w_{\beta j}\) such that \(x\in w_{\beta j}\). Let us verify that

\[ x\in w_{\alpha i}\cap w_{\beta j}\subseteq Ox. \]

* A mapping is finitely multiple if the preimage of every point consists of a finite number of points. A mapping is \(k\)-multiple if the preimage of every point consists of exactly \(k\) points.

** Everywhere in this paper, by a space is meant a Hausdorff space, and by a mapping—a continuous mapping.

*** A mapping is compact if the preimage of every point is compact.

**** Obviously, the weight is not increased either.

If it were the case that \((w_{\alpha i}\setminus Ox)\cap w_{\beta j}\ne \Lambda\), then we would have \(v_\beta\cap f(w_{\alpha i}\setminus Ox)\ne \Lambda\), but this is not so. Hence \((w_{\alpha i}\setminus Ox)\cap w_{\beta j}=\Lambda\), whence it follows that \(w_{\alpha i}\cap w_{\beta j}\subseteq Ox\). The lemma is proved.

We shall now prove the theorem without difficulty. There is an open finite-to-one mapping \(f:X\to Y\), where \(X\) is locally bicompact; \(Y=\bigcup_{k=1}^{\infty}Y_k\), where \(Y_k\) consists only of points of multiplicity \(k\) of the mapping \(f\). Correspondingly \(X=\bigcup_{k=1}^{\infty}X_k\), where \(X_k=f^{-1}Y_k\); the mapping \(f:X_k\to Y_k\) is open and \(k\)-to-one. In view of the lemma, the weight of \(X_k\leq\) the weight of \(Y_k\). Applying the known sum theorem from \((^1)\), we obtain that the weight of \(X\leq\) the weight of \(Y\).

Theorem \(1'\). Let there be a finite-to-one open mapping \(f:X\to Y\), where \(X\) is an \(m\)-compact space, and the weight of \(Y\leq m\). Then the weight of \(X\leq m\).

Corollary. If a bicompactum is mapped finite-to-one and openly onto a compactum, then such a bicompactum is metrizable.

In connection with these facts there arises a question, the answer to which is unknown: does there exist a nonmetrizable bicompactum which is mapped openly and compactly onto a compactum?

The following proposition gives a partial answer to this question (and even to a stronger one).

Proposition 1. If a dyadic* bicompactum \(X\) is mapped compactly onto a compactum, then the bicompactum \(X\) is metrizable.

Indeed, it is easy to show that the bicompactum \(X\) will satisfy the first axiom of countability, and then, by a known theorem \((^2)\), it is metrizable.

In the proof of Theorem 1 only the fact was used that the sum theorem is valid for a locally bicompact space. Therefore the analogous assertion will be true for any space in which the sum theorem holds. In view of this the following propositions are valid.

Theorem \(1''\). Under a finite-to-one open mapping, the weight of a \(p\)-space \((^7)\) is not lowered.

Corollary 1. Under a finite-to-one open mapping, the weight of a metric space is not lowered.

Corollary 2. Under a finite-to-one open mapping, the weight of a space complete in the sense of Čech is not lowered.

Theorem \(1'''\). Under a finite-to-one open mapping, the weight of a space having a perfectly normal bicompact extension is not lowered.

Next we give facts concerning \(k\)-to-one open mappings.

Theorem 2. Let there be an open \(k\)-to-one mapping \(f:X\to Y\), where \(Y\) is a paracompact space. Then \(X\) is also a paracompact space.

Proof. Consider an arbitrary covering \(\omega=\{\omega_\alpha\}\) of the space \(X\); inscribe in the covering \(f\omega=\{f\omega_\alpha\}\) of the space \(Y\) a special covering \(u=\{u_\alpha\}\) as follows. For every point \(y\in Y\) there will be found a neighborhood \(u_\alpha\ni y\) such that
\[ f^{-1}u_\alpha=\bigcup_{i=1}^{k}v_{\alpha i}, \]
where each \(v_{\alpha i}\) is open and is contained entirely in one of the elements of \(\omega\), and, moreover, \(fv_{\alpha i}=u_\alpha,\ i=1,\ldots,k\). The covering \(u=\{u_\alpha\}\) is inscribed in \(f\omega=\{f\omega_\alpha\}\), and the covering \(v=\{v_{\alpha i}\}\) is inscribed in \(\omega=\{\omega_\alpha\}\).

* A bicompactum homeomorphic to \(D^\tau\) (the product of \(\tau\) copies of the two-point space) is called dyadic. Every dense subspace of a dyadic bicompactum is called a dyadic space.

We inscribe in the cover \(u=\{u_\alpha\}\) a locally finite cover \(w=\{w_\alpha\}\); for every \(w_\alpha\in w\) we choose some one \(u_\beta\in u\) such that \(w_\alpha\subseteq u_\beta\), and put \(\theta_{\alpha i}=f^{-1}w_\alpha\cap v_{\beta i}\), \(i=1,\ldots,k\). The cover \(\theta=\{\theta_{\alpha i}\}\) is a locally finite cover inscribed in \(\omega\).

Corollary. If a space \(X\) is open and \(k\)-to-one mapped onto a metrizable space, then it is itself metrizable.

The space \(X\) is locally metrizable, since an open \(k\)-to-one mapping is a local homeomorphism. From our Theorem 2 it follows that \(X\) is paracompact. But a locally metrizable paracompact space is metrizable \((^4)\).

Theorem 3. If a space \(Y\) is an open \(k\)-to-one image of a paracompact space, then \(Y\) itself is paracompact.

Let \(\omega=\{\omega_\alpha\}\) be an arbitrary cover of the space \(Y\); in it, as we have seen, one can inscribe a cover \(u=\{u_\alpha\}\) such that

\[ f^{-1}u_\alpha=\bigcup_{i=1}^{k} v_{\alpha i},\qquad fv_{\alpha i}=u_\alpha,\quad i=1,\ldots,k, \]

so that on each \(v_{\alpha i}\) the mapping \(f\) is topological. In the cover \(v=\{v_{\alpha i}\}\) of the space \(X\) we inscribe a locally finite \(w=\{w_\alpha\}\). It is possible to verify that \(fw=\{fw_\alpha\}\) is a locally finite cover inscribed in \(\omega\). Indeed, let \(y\in Y\) be an arbitrary point of \(Y\); we indicate for it a neighborhood which intersects only finitely many elements of the cover \(fw\);

\[ f^{-1}y=\bigcup_{i=1}^{k} x_i . \]

For each point \(x_i\) we take a neighborhood \(Ox_i\) such that it intersects only finitely many elements of \(w\). The required neighborhood is

\[ Oy=\bigcap_{i=1}^{k} fOx_i . \]

The assertion is proved.

Corollary. If a space \(Y\) is an open \(k\)-to-one image of a metrizable space, then \(Y\) itself is metrizable.

Theorem \(2'\). Suppose there is an open \(k\)-to-one mapping \(f:X\to Y\), where \(Y\) is a weakly paracompact (strongly paracompact) space. Then \(X\) is also weakly paracompact (respectively, strongly paracompact).

Theorem \(3'\). If a space \(Y\) is an open \(k\)-to-one image of a weakly paracompact space, then \(Y\) itself is weakly paracompact.

The methods of proof of the last theorems are the same as for Theorems 2 and 3.

Example 1. An example of an open finite-to-one mapping of a nonparacompact space onto a paracompact space (showing that Theorem 2 is false for finite-to-one mappings). As the space \(Y\) one takes the space \(T\) of all ordinal numbers up to \(\omega_1\), inclusive. The space \(X\) consists of two isolated copies of the space \(T\), one of which is taken without the point \(\omega_1\). The mapping \(f\) of the space \(X\) onto \(Y\) is defined as follows: to each ordinal number \(x\in X\) there is assigned the equal ordinal number in \(Y\). The mapping \(f\) is open and two-valued (but not 2-to-one!).

Example 2. An example of an open finite-to-one mapping of a nonmetrizable normal space onto a compactum. To the Gol'tsin space \((^5)\), along the boundary circle, an ordinary disk is glued—thus we obtain the space \(X\). The space \(X\), obviously, two-valuedly and openly “projects” onto the ordinary disk.

Moscow State University
named after M. V. Lomonosov

Received
28 IV 1965

REFERENCES

\(^{1}\) A. Arkhangel’skii, DAN, 126, No. 2, 239 (1959).
\(^{2}\) A. Esenin-Vol’pin, DAN, 68, No. 3, 649 (1949).
\(^{3}\) B. Efimov, DAN, 151, No. 5, 1021 (1963).
\(^{4}\) Yu. M. Smirnov, UMN, No. 6, 100 (1951).
\(^{5}\) A. Archangelski, W. Holsztynski, Bull. Acad. Polon. Sci., 11, No. 8, 493 (1963).
\(^{6}\) V. Proizvolov, DAN, 159, No. 3, 516 (1964).
\(^{7}\) A. Arkhangel’skii, DAN, 151, No. 4, 751 (1963).