UDC 513.831

MATHEMATICS

N. V. VELICHKO

AN EXAMPLE OF AN OPEN COMPACT MONOTONE CONTINUOUS MAPPING OF A NONMETRIZABLE BICOMPACTUM ONTO A COMPACTUM

(Presented by Academician P. S. Aleksandrov, 27 II 1967)

V. V. Proizvolov posed the following problem. Let \(f:X\to Y\) be an open compact mapping, \(X\) a bicompactum, \(Y\) a compactum. Must the space \(X\) be metrizable? (see \((^1)\)).

We shall call a mapping \(f:X\to Y\) compact if the preimage \(f^{-1}y\) is compact \((=\) a metrizable compact set) for every point \(y\in Y\). A mapping \(f\) is called monotone if all \(f^{-1}y\) are connected. We construct an example that gives a negative solution of the problem.

In the numerical plane \(R^2\) consider the square
\(I=E\{(x,y):-1\le x\le 1,\ -1\le y\le 1\}\). Denote by \(P\) the set \(E\{(x,y):y\le 0\}\). Let
\(S=I\setminus P\), \(T=E\{(x,y):y=-1\}\), \(G=E\{(x,y):y=0\}\).
On the set \(I\) consider the following topology \(t\). At the points of \(S\) the topology induced on \(S\) by the topology of the numerical plane \(R^2\) is preserved.

Let \(p=(x,y)\in P\setminus (T\cup G)\) be an arbitrary point, \(p_0=(x,0)\). In \(S\) construct two rays \(l_p\) and \(\bar l_p\), issuing from the point \(p_0\), symmetric with respect to the line \(l_{p_0}\) passing through the point \(p_0\) parallel to the axis \(OY\), and such that the angle between \(l_p\) and \(l_{p_0}\) is equal to \(|y|d\), where \(d\) is a right angle. Let \(q\) be an arbitrary point on \(l_p\), and let \(\bar q\) be the point symmetric to it with respect to \(l_{p_0}\) (if \(\bar q\notin S\), then for \(\bar q\) we take the point of intersection of the line \(\bar l_p\) with the side of the square). Denote by \(m\) and \(\bar m\) the lines in \(S\) perpendicular to \(l_p\) and \(\bar l_p\) at the points \(q\) and \(\bar q\).

Let \(\varepsilon\) be a positive number,
\(\varepsilon\le \min |y|,(1-|y|)\). Introduce the notation:
\(E_{\varepsilon p}=E\{(x',y'):x'=x,\ |y-y'|<\varepsilon\}\),
\(p_1=(x,y+\varepsilon)\), \(p_2=(x,y-\varepsilon)\). Further, let \(q_{p_1}\) and \(q_{p_2}\), \(\bar q_{p_1}\) and \(\bar q_{p_2}\) be the points of intersection of the lines \(l_{p_1}, l_{p_2}\) and \(\bar l_{p_1}, \bar l_{p_2}\) (which are constructed for the points \(p_1\) and \(p_2\) in the same way as the lines \(l_p\) and \(\bar l_p\) for the point \(p\)) with the lines \(m\) and \(\bar m\). Denote by \(Q_{mp}\) the set consisting of the interior points of the two triangles \((p_0,q_{p_1},q_{p_2})\) and \((p_0,\bar q_{p_1},\bar q_{p_2})\).

Put \(O_{\varepsilon mp}=E_{\varepsilon p}\cup Q_{mp}\). By definition, the basis of the system of neighborhoods of the point \(p\) in the new space \(\nabla=(I,t)\) will consist of sets of the type \(O_{\varepsilon mp}\) for all possible
\(\varepsilon\le \min |y|,(1-|y|)\) and for arbitrarily chosen points \(q\) on \(l_p\), determining the lines \(m\).

If the point \(p=p_0\in G\), then \(l_p=\bar l_p=l_{p_0}\), \(m\) is a line in \(S\) perpendicular to \(l_{p_0}\) at some point \(q\), \(\varepsilon\le 1\), \(p_1=p_0\), \(Q_{mp_0}\) consists of the interior points of one triangle \((p_0,q_{p_2},\bar q_{p_2})\),
\(E_{\varepsilon p_0}=E\{(x',y'):x=x',\ |y'|<\varepsilon,\ y'\le 0\}\).
Sets of the type
\(O_{\varepsilon mp_0}=E_{\varepsilon p_0}\cup Q_{mp_0}\)
form, by definition, the basis of the system of neighborhoods of the point \(p_0\).

Finally, let the point \(d=(x,-1)\in T\). Here \(\varepsilon\le 1\), \(p_2=d\), and \(l_d\) and \(\bar l_d\) lie on the segment \(G\); \(m\) and \(\bar m\) are lines parallel to the axis \(OY\), meeting \(G\) at the points \(q=(z,0)\) and \(\bar q=(\bar z,0)\). The set \(Q_{md}\) consists of the interior points of the two triangles \((p_0,q,q_{p_1})\) and \((p_0,\bar q,\bar q_{p_1})\),
\(E_{\varepsilon d}=E\{(x',y'):x'=x,\ 1-|y'|<\varepsilon,\ y'\ge -1\}\).
Sets of the type
\(O_{\varepsilon md}=E_{\varepsilon d}\cup Q_{md}\)
do not define here a basis of the system of neighborhoods.

Put
\(V_{md}=I\{(x',y'):(x',y')\in P,\ z<x'<\bar z,\ x'\ne x\}\).
And only sets of the type
\(W_{\varepsilon md}=O_{\varepsilon md}\cup V_{md}\)
will constitute the basis of the system of neighborhoods of the point \(d\in T\).

Thus, a new topological space \(\nabla\) has been constructed (the correctness of its definition is easy to verify). We shall prove that \(\nabla\) is a nonmetrizable bicompactum. It is easy to establish that the space \(\nabla\) is Hausdorff.

Let \(M\) be an arbitrary infinite set lying in \(P\), and let \(p=(x,y)\) be a point of complete accumulation of \(M\) in the bicompactum \(I\). If \(p\) is not a point of complete accumulation of \(M\) in the space \(\nabla\), then the point \(d=(x,-1)\) will be a point of complete accumulation of \(M\) in \(\nabla\), as follows easily from the construction of the topology at the point \(d\). Hence it follows that the space \(P\subset\nabla\) is bicompact.

(A). Let the point \(z=(x,0)\in G\), and
\[ L=E\mid (x',y'):\ x'=x,\ -1\leq y\leq 0\mid . \]
If, for each point \(p\in L\), one arbitrarily chooses a basic neighborhood \(O_{em}p\), then the set
\[ V_z=\bigcup_{p\in L}(O_{em}p\cap K) \]
will be a neighborhood of the point \(z\) in the rectangle \(K=S\cup G\) (in its natural topology). Indeed, projecting from the point \(z\) each \(O_{em}p\) onto the closed set
\[ C=E\mid (x',y'):\ y'=1\mid \cup E\mid (x',y'):\ x'=-1,\ 1,\ y\geq 0\mid , \]
we obtain an open cover of the bicompactum \(C\). From this cover choose a finite subcover, to each element of which there corresponds some \(O_{em}p_i\), \(i=1,2,\ldots,n\), projecting into it. Let \(j\) be such that the segment \(l_{p_j}\cap O_{em}p_j\) has the smallest length \(d\) (among the segments \(l_{p_i}\cap O_{em}p_i\), \(i=1,2,\ldots,n\)). Then the open disk (in \(K\)) with center at the point \(z\) and radius \(d\) will be contained in \(V_z\subset K\). It follows from what has been proved that, if the open (in \(K\)) sets \(V_z\) are chosen for each point \(z\in G\), then the set
\[ \bigcup_{z\in G} V_z \]
will be a neighborhood of the segment \(G\) in the bicompactum \(K\).

Now let \(\Sigma=\{H\}\) be an arbitrary open cover of the space \(\nabla\). There exists a finite number of sets \(H_i\in\Sigma\), \(i=1,2,\ldots,n\), covering \(P\):
\[ P\subseteq \bigcup_{i=1}^n H_i . \]
From the assertion proved in (A) it follows that the set
\[ A=\left(\bigcup_{i=1}^n H_i\right)\cap K \]
will be an open neighborhood of the set \(G\) in \(K\); therefore the set \(B=K\setminus A\) is closed and hence bicompact in \(K\), and since it lies in \(S\), it is bicompact in \(\nabla\). There is a finite number of sets \(H_j\), \(j=1,2,\ldots,m\), such that
\[ B\subseteq \bigcup_{j=1}^m H_j . \]
But then
\[ \nabla=\bigcup_{i=1}^n H_i\cup \bigcup_{j=1}^m H_j . \]
It is proved that \(\nabla\) is a bicompact space.

The space \(\nabla\) is not metrizable. Its subspace \(S\cup G\) is completely regular, but not normal. This can be proved by the same method by which the nonnormality of the space of V. V. Nemytzkiĭ is proved (see, for example, \((^2)\)).

“Project” the bicompactum \(\nabla\) onto the upper side
\[ \Delta=E\mid (x,y):\ y=1\mid \]
of the square \(I\), i.e., assign to the point \((x,y)\in\nabla\) the point \((x,1)\). This projection \(\varphi\) will be a continuous mapping of \(\nabla\) onto the compactum \(\Delta\). The preimage of the point \((x,1)\) under this mapping will be the set
\[ E\mid (x',y):\ x'=x\mid , \]
homeomorphic to the segment \([0,1]\). Consequently, \(\varphi\) is a compact monotone mapping. It is easy to verify that it is open. The example is constructed.

Let us call a bicompactum that is mapped openly and compactly onto a compactum a bicompactum of type \(S\). The following are easily proved:

Theorem 1. A countable product of bicompacta of type \(S\) is a bicompactum of type \(S\).

Theorem 2. A topological (discrete) sum of bicompacta of type \(S\) is a space that is perfectly openly and compactly mapped onto a metrizable space.

Ural State University
named after A. M. Gorky

Received
1 II 1967

REFERENCES

1. V. V. Proizvolov, DAN, 166, 38 (1966).
2. Yu. M. Smirnov, Uchen. zap. MGU, 155, 137 (1952).