UDC 513.83

MATHEMATICS

V. V. Filippov

ON THE PRESERVATION OF THE MULTIPLICITY OF A BASE UNDER A PERFECT MAPPING

(Presented by Academician P. S. Aleksandrov on XII 1, 1967)

In the present note a positive answer is given to the question posed by P. S. Aleksandrov: is the multiplicity of a base preserved under a perfect mapping?

Definition*. We say that the multiplicity of a family \(\Phi\) of subsets of a set \(F\) does not exceed the cardinal number \(\tau\), if each point of the set \(F\) is contained in no more than \(\tau\) elements of the family \(\Phi\).

In fact it turned out that even the following is true.

Theorem 1. Let \(f:X\to Y\) be a pseudo-open bicompact mapping of a \(T_1\) space \(X\), possessing a base of multiplicity \(\leq\tau\), onto the space \(Y\). Then \(Y\) also possesses a base of multiplicity \(\leq\tau\)*.

Corollary. Let \(f:X\to Y\) be a perfect mapping, and let \(X\) have a base of multiplicity \(\leq\tau\). Then \(Y\) also has a base of multiplicity \(\leq\tau\).

This is precisely the answer to P. S. Aleksandrov’s question mentioned above.

In the present note we shall give a direct proof of the corollary and an outline of the proof of Theorem 1.

We begin the proof of the corollary with the following lemma.

Lemma. Let \(Z'\subseteq Z\) be arbitrary sets, and let \(\theta'\), \(\theta\) be families of subsets, respectively of \(Z'\) and \(Z\), whose multiplicity does not exceed \(\tau\). Consider the sets

\[ \gamma_{\vartheta'}=\bigcup\{\vartheta,\ \vartheta\in\theta,\ \varnothing\ne\vartheta\cap Z'\subseteq\vartheta'\}. \]

Under the assumptions made, the multiplicity of the family \(\Gamma=\{\gamma_{\vartheta'}\}_{\vartheta'\in\theta'}\) does not exceed \(\tau\).

First of all, the set \(\vartheta\cap Z'\) cannot be contained in more than \(\tau\) elements of the family \(\theta'\), since the multiplicity of \(\theta'\) is not greater than \(\tau\). Therefore \(\vartheta\in\theta\) occurs as a summand in no more than \(\tau\) sets \(\gamma_{\vartheta'}\). A point \(z\in Z\) is contained only in those elements \(\gamma_{\vartheta'}\in\Gamma\) for which there is a \(\vartheta\in\theta\), contained in \(\gamma_{\vartheta'}\in\Gamma\) as a summand and containing \(z\); and, by what has been said above, there are no more than \(\tau^2=\tau\) such elements, as was required.

Let \(B^X\) be a fixed base of the space \(X\), whose multiplicity is \(\leq\tau\), and let \(B_n^X\) \((n=1,2,\ldots)\) be the set of families of elements of the base \(B^X\)

* All other definitions, as well as the necessary comments, may be found in (1).

** Here, and everywhere below, although this is not specially stated, we require that the image space be a \(T_1\)-space, so that A. S. Mishchenko’s theorem (2) may be used. All mappings are assumed to be mappings onto.

* Note added in proof. After submitting the article to the editors the author obtained a stronger result:

Theorem. Let \(f:X\to Y\) be a bifactorial \(\tau\)-mapping of a \(T_1\)-space \(X\), possessing a base of multiplicity \(\leq\tau\), onto a space \(Y\). Then \(Y\) also possesses a base of multiplicity \(\leq\tau\).

A mapping \(f:X\to Y\) is called bifactorial if from every cover \(\{U_\alpha\}\) of the preimage \(f^{-1}(y)\) of any point \(y\in Y\) one can choose a finite number of elements \(\{U_1,\ldots,U_n\}\) such that \(y\in \operatorname{Int} f(U_1\cup\cdots\cup U_n)\), and it is called a \(\tau\)-mapping if the weight of the preimage of any point \(y\in Y\) does not exceed \(\tau\).

\(\beta=\{b_1,\ldots,b_n\}\), for which there exists a point \(y\in Y\) such that the sets \(b_1,\ldots,b_n\) form a minimal covering* of its inverse image \(f^{-1}(y)\). The inverse image of each such point \(y\in Y\) we shall call an essential element of the family \(\beta\). By induction on \(n\) and on all possible triples \(f:X\to Y\), where the space \(X\) has a base \(B^X\) of multiplicity \(\leqslant\tau\), and the mapping \(f\) is perfect, we shall prove that there exists a family \(\Phi_n^X\) of multiplicity \(\leqslant\tau\) of open subsets of the space \(X\) and a one-to-one correspondence \(g_n^X:B_n^X\to\Phi_n^X\) such that
\[ g_n^X(\beta)\subseteq \bigcup_{b\in\beta} b \]
and \(g_n^X(\beta)\) contains all essential elements of \(\beta\). (Here, incidentally, a situation may arise in which two elements \(\varphi_1,\varphi_2\in\Phi_n^X\) coincide as sets lying in the space \(X\), but, being put into correspondence with different elements of the family \(B_n^X\), they will differ as elements of \(\Phi_n^X\).)

As \(\Phi_1^X\) we take \(B_1^X\) and as \(g_1^X\) the identity mapping. The multiplicity of \(B_1^X\leqslant\tau\), since \(B_1^X\subseteq B^X\) and the multiplicity of \(B^X\leqslant\tau\).

Suppose the assertion has been proved for any space \(X\) and \(n=k\). We prove the assertion for \(n=k+1\). Take an arbitrary \(b\in B^X\). Consider \(X'=X\setminus b\). As \(B^{X'}\) we take the intersections of the elements of \(B^X\) with \(X'\). By the induction hypothesis** for \(X'\) there has been constructed a family \(\Phi_k^{X'}\) and a correspondence \(g_k^{X'}\). To each set \(\varphi'\in\Phi_k^{X'}\) assign the set
\[ \varphi=f^{-1}f(X')\setminus f^{-1}f(X'\setminus\varphi'). \]
We apply the lemma, putting \(\theta'=\{\varphi\}\), \(\theta=B^X\), \(Z'=f^{-1}f(X')\), \(Z=X\). From A. S. Mishchenko’s theorem \((^2)\) it follows easily that the multiplicity of the family \(\{\varphi\}\) does not exceed \(\tau\), and hence we are in the conditions of the lemma. The family \(\{\gamma_\varphi\}=\Gamma_{k+1}^b\) constructed in accordance with the lemma has multiplicity \(\leqslant\tau\). As a sum of open sets, each element \(\Gamma_{k+1}^b\) is open. Let \(\Delta_{k+1}^b\) be the part of \(B_{k+1}^X\) consisting of the families containing the set \(b\). Define the correspondence
\[ g_{k+1}^b:\Delta_{k+1}^b\to\Gamma_{k+1}^b \]
as follows:
\[ g_{k+1}^b(\{b,b_1,\ldots,b_k\}) = \gamma_{\,f^{-1}f(X')\setminus f^{-1}f\bigl(X'\setminus g_k^{X'}(\{b_1\cap X',\ldots,b_k\cap X'\})\bigr)}. \]

Let us note some properties of the correspondence \(g_{k+1}^b\). \(g_{k+1}^b(\beta)\subseteq\bigcup_{b\in\beta} b\)—this follows from the inclusions
\[ \varphi=f^{-1}f(X')\setminus f^{-1}f(X'\setminus\varphi')\subseteq b\cup\varphi', \]
where
\[ \varphi'=g_k^{X'}(\{b_1\cap X',\ldots,b_k\cap X'\})\subseteq \bigcup_{i=1}^{k}(b_i\cap X')\subseteq\bigcup_{i=1}^{k} b_i, \]
\[ \gamma_\varphi\subseteq (X\setminus f^{-1}f(X'))\cup\varphi\subseteq b\cup\varphi\subseteq b\cup\varphi'\subseteq b\cup b_1\cup\cdots\cup b_k. \]
We show that \(g_{k+1}^b(\beta)\) contains all essential elements of \(\beta\). If \(y\in Y\) and \(f^{-1}(y)\) is an essential element of \(\{b,b_1,\ldots,b_k\}\), then \(f^{-1}(y)\setminus b\), evidently, is an essential element of \(\{b_1\cap X',\ldots,b_k\cap X'\}\) (for \(f|_{X'}\), the restriction of the mapping \(f\) to \(X'\)), and therefore
\[ f^{-1}(y)\setminus b\subseteq g_k^X(\{b_1\cap X',\ldots,b_k\cap X'\}). \]
The set
\[ X'\setminus g_k^X(\{b_1\cap X',\ldots,b_k\cap X'\}) \]
is closed in \(X\), and therefore the set
\[ f^{-1}f\bigl(X'\setminus g_k^X(\{b_1\cap X',\ldots,b_k\cap X'\})\bigr) \]
is also closed in \(X\) and does not meet \(f^{-1}(y)\). For each point \(x\in f^{-1}(y)\) there is found a base element not meeting the set
\[ f^{-1}f\bigl(X'\setminus g_k^{X'}(\{b_1\cap X',\ldots,b_k\cap X'\})\bigr). \]
All such elements will enter into
\[ \gamma_{\,f^{-1}f(X')\setminus f^{-1}f\bigl(X'\setminus g_k^{X'}(\{b_1\cap X',\ldots,b_k\cap X'\})\bigr)} \]
as summands, whence it follows that
\[ f^{-1}(y)\subseteq g_{k+1}^b(\{b,b_1,\ldots,b_k\}). \]

We construct the family \(\Phi_{k+1}^X\) in the following way. Let \(\{b_1,\ldots,b_{k+1}\}\in B_{k+1}^X\); then
\[ g_{k+1}^X(\{b_1,\ldots,b_{k+1}\}) = \bigcup_{i=1}^{k+1} g_{k+1}^{b_i}(\{b_1,\ldots,b_{k+1}\}). \]
Let, further, \(\Phi_{k+1}^X=\)

* A family of sets \(\{m_1,\ldots,m_n\}\) is called a minimal covering of a set \(M\) if
\[ M\subseteq\bigcup_{i=1}^{n} m_i \]
and
\[ M\nsubseteq\bigcup_{j=1}^{l} m_{i_j} \]
for any proper subfamily
\[ \{m_{i_1},\ldots,m_{i_l}\}\subset\{m_1,\ldots,m_n\}. \]

** As is easily seen, the restriction of a perfect mapping to a closed subspace is perfect.

\[ =\{g_{k+1}^X(\beta),\ \beta\in B_{k+1}^X\}. \]
By the properties of \(g_{k+1}^b\) and the definition of \(g_{k+1}^X\) we have
\[ g_{k+1}^X(\beta)\subseteq \bigcup_{b\in\beta} b, \]
and \(g_{k+1}^X(\beta)\) contains all essential elements of \(\beta\). We show that the multiplicity of \(\Phi_{k+1}^X\) is \(\leq \tau\). Let
\[ \Phi_{k+1}^X(b)=\{g_{k+1}^X(\beta),\ \beta\ni b\}. \]
Since the family \(\Phi_{k+1}^X(b)\) is elementwise* inscribed in \(\Gamma_{k+1}^b\), the multiplicity of \(\Phi_{k+1}^X(b)\) is \(\leq \tau\). It is easy to see that the family
\[ \bigcup_{b\ni x}\Phi_{k+1}^X(b) \]
exhausts all elements of \(\Phi_{k+1}^X\) containing the point \(x\in X\). Each of the families \(\Phi_{k+1}^X(b)\) has multiplicity \(\leq\tau\), and there are altogether \(\leq\tau\) such families; consequently the multiplicity
\[ \Phi_{k+1}^X\leq \tau^2=\tau. \]
Let
\[ \Phi^X=\bigcup_{n=1}^{\infty}\Phi_n^X. \]
We shall not give the verification that the family
\[ \{Y\setminus f(X\setminus \varphi),\ \varphi\in\Phi^X\} \]
is a certain base of the space \(Y\) and that its multiplicity is \(\leq\tau\). In doing this one must use the fact that the mapping \(f\) is perfect. This completes the proof of the corollary.

We now give the plan of the proof of Theorem 1.

I. Let \(Y'\subseteq Y,\ X'=f^{-1}(Y')\); then \(f([X'])=[f(X')]\).

II. If \(Y'\subseteq Y,\ X'=f^{-1}(Y')\), and \(G=\{g\}\) is a family of multiplicity \(\leq\tau\) of open subsets of \(X\), then there exists a family \(H=\{h\}\) of multiplicity \(\leq\tau\) of open subsets of \(X\) and a correspondence \(\eta:G\to H\) such that

a) \(\eta(g)\cap X'\subseteq g\);

b) if \(y\in Y\), \(f^{-1}(y)\cap [X']\ne\varnothing\), and
\[ f^{-1}(y)\cap [X'\setminus g]=\varnothing, \]
then
\[ \eta(g)\supseteq f^{-1}(y). \]

III. Let \(B\) be a base of the space \(X\) of multiplicity \(\leq\tau\),
\[ B_1=\{f^{-1}f(b),\ b\in B\}, \]
\[ B_n=\{\{b_1,\ldots,b_n\},\ b_1,\ldots,b_n\in B_1\}, \]
and \(X'=f^{-1}(Y')\), where \(Y'\subseteq Y\). Let, for
\[ \beta=\{b_1,\ldots,b_n\}\in B_n, \]
\[ q(X',\beta)\cup \{f^{-1}(y),\ y\in Y,\ f^{-1}(y)\cap [X']\ne\varnothing,\ f^{-1}(y)\cap [X'\setminus \bigcup_{b\in\beta} b]=\varnothing \]
and for every proper subfamily \(\{b_{i_1},\ldots,b_{i_p}\}\) of the family \(\{b_1,\ldots,b_n\}\),
\[ [X'\setminus (b_{i_1}\cup\cdots\cup b_{i_p})]\cap f^{-1}(y)\ne\varnothing\}. \]
Then by induction on \(n\) it is proved that for every \(n\) there exists a family \(\Phi_n^{X'}\) of multiplicity \(\leq\tau\) of open subsets of the space \(X\) and a correspondence
\[ g_n^{X'}:B_n\to \Phi_n^{X'} \]
such that
\[ g_n^{X'}\supseteq q(X',\beta) \]
and
\[ g_n^{X'}(\beta)\cap \left(X'\setminus\left(\bigcup_{b\in\beta} b\right)\right)=\varnothing. \]

IV. Let
\[ \Phi=\bigcup_{n=1}^{\infty}\Phi_n^X. \]
Then the family
\[ \{Y\setminus f(X\setminus \varphi),\ \varphi\in\Phi\} \]
is a certain base of the space \(Y\), whose multiplicity is \(\leq\tau\).

By methods analogous to those developed in the proof of Theorem 1 and its corollary, one can obtain the following results.

Theorem 2. Let \(f:X\to Y\) be a perfect mapping, and let \(X\) possess a base that decomposes into \(\tau\) point-finite families; then \(Y\) also possesses a base that decomposes into \(\tau\) point-finite families.

Definition. A base that decomposes into \(\tau\) locally finite families will be called an \(NS\tau\)-base.

Theorem 3. Let \(f:X\to Y\) be a perfect mapping, and let \(X\) possess an \(NS\tau\)-base; then \(Y\) also possesses an \(NS\tau\)-base.

In particular, for \(\tau=\aleph_0\), under the assumption that \(X\) is normal, we obtain A. H. Stone’s theorem \((^3)\), which states that the image of a metrizable space under a perfect mapping is metrizable.

I express my deep gratitude to A. V. Arhangel’skii, whose advice I continually used.

Moscow State University
named after M. V. Lomonosov

Received
1 XII 1967

References

1. A. V. Arhangel’skii, Uspekhi Mat. Nauk, 21, no. 4, 133 (1966).
2. A. S. Mishchenko, DAN, 144, 985 (1962).
3. A. H. Stone, Proc. Am. Math. Soc., 7, 690 (1956).

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* The elements of the families \(\Phi_{k+1}^X(b)\) and \(\Gamma_{k+1}^b\) are numbered by the set \(\Delta_{k+1}^b\). Elements \(\varphi\in\Phi_{k+1}^X(b)\) and \(\gamma\in\Gamma_{k+1}^b\) that correspond to one and the same element of \(\Delta_{k+1}^b\) satisfy the relation \(\varphi\subseteq\gamma\).