A. D. TAIMANOV

EXTENSION OF MONOTONE MAPPINGS TO MONOTONE MAPPINGS OF BICOMPACTA

(Presented by Academician P. S. Aleksandrov on 11 VI 1960)

At the Second All-Union Topological Conference V. Ponomarev posed the following question: is it always possible to extend a monotone mapping \(f\) of a space \(X\) onto a space \(Y\) to a monotone mapping of some extension* \(bX\) onto some extension \(bY\)? Here a positive solution of V. Ponomarev’s problem is given for all closed and all open mappings in the case where the extension of the space \(Y\) is the Čech (maximal) extension \(\beta Y\). At the end two examples proposed by Yu. M. Smirnov are given. The first shows that even for identity mappings the condition of maximality of the extension \(bY\) (i.e. that \(bY=\beta Y\)) is essential, and the second—that for arbitrary mappings the theorem is false.

Theorem. Let \(bX\) be an arbitrary extension of a completely regular space \(X\), let \(\beta Y\) be the Čech extension of a normal space \(Y\), and let \(f_b\) be a mapping of the bicompactum \(bX\) onto \(\beta Y\) which is the (unique) extension of a monotone mapping \(f\) of the space \(X\) onto the space \(Y\); if \(f\) is open or closed, then the extension \(f_b\) is also monotone.

Proof**. We make the following simple observation.

Remark. Suppose a mapping \(f\) of a space \(X\) into a space \(Y\) is given; if its extension \(f_\beta\), carrying \(\beta X\) into \(\beta Y\), is monotone, then for every such extension \(bX\) of the space \(X\) for which the mapping \(f\) can be extended to a mapping \(f_b\) of the extension \(bX\) into \(\beta Y\), this extension \(f_b\) is also monotone.

Indeed, under the conditions of the remark, for any such extension \(bX\) we have

\[ f_\beta=f_b\vartheta, \]

where \(\vartheta\) is the mapping of the bicompactum \(\beta X\) onto \(bX\) that is identical on \(X\). Therefore, if for some point \(y\) of \(\beta Y\) the preimage*** \(f_b^{-1}y\) were disconnected, then the preimage
\[ f_\beta^{-1}y=\vartheta^{-1}f_b^{-1}y \]
would also be disconnected.

In view of this remark it is enough to carry out the

Proof of the theorem for the extension \(f_\beta\). Suppose the hypotheses of the theorem are satisfied and at the same time the preimage \(f_\beta^{-1}y\) of some point \(y\in\beta Y\) is disconnected, i.e.
\[ f_\beta^{-1}y=A\cup B, \]
where \(A\) and \(B\) are nonempty, closed, and disjoint. By the normality of the bicompactum \(\beta X\) there exist

* By an extension we shall everywhere mean only bicompact extensions, and by mappings—continuous mappings.

** My original proof was somewhat simplified by V. T. Levshenko and Yu. M. Smirnov. I sincerely thank them.

*** Everywhere only complete preimages are considered.

there exist neighborhoods \(U\) and \(V\) of the set \(A\), respectively of the set \(B\), such that \(\overline U^{\beta}\cap \overline V^{\beta}=\varnothing\)*. Since \(A\subseteq \overline U^{\beta}=\overline{U\cap X}^{\beta}\), it follows that \(y\in \overline{f(U\cap X)}^{\beta}\). Similarly, \(y\in \overline{f(V\cap X)}^{\beta}\). Therefore

\[ y\in \overline{f(U\cap X)}^{\beta}\cap \overline{f(V\cap X)}^{\beta}. \tag{1} \]

By the normality of the space \(Y\), the closure operator in \(\beta Y\), applied to closed subsets of \(Y\), commutes with the operation of intersection (¹). Therefore

\[ y\in \overline{\,f(U\cap X)\cap \overline{f(V\cap X)}\,}^{\beta}. \tag{2} \]

Since \(f_\beta^{-1}y\subset U\cup V\) and the mapping \(f_\beta\) is closed, there exists a neighborhood \(Oy\) such that

\[ f_\beta^{-1}(Oy)\subseteq U\cup V. \tag{3} \]

First case: \(y\in Y\). Then, by connectedness of the set \(f^{-1}y\), either \(f^{-1}y\subseteq U\), or \(f^{-1}y\subseteq V\); suppose, for example, that \(f^{-1}y\subseteq U\). Hence, on the basis of (1), we have

\[ y\in Oy\cap f(U\cap X)\cap \overline{f(V\cap X)}. \]

If the mapping \(f\) is open, then the image \(f(U\cap X)\) is open in \(Y\), and therefore the set \(H=Oy\cap f(U\cap X)\cap \overline{f(V\cap X)}\) is nonempty. Let \(y'\in H\). Then
\(f^{-1}y'\subseteq f_\beta^{-1}(Oy)\subseteq U\cup V\), \(f^{-1}(y')\cap U\ne\varnothing\) and \(f^{-1}(y')\cap V\ne\varnothing\), which is impossible. If, however, the mapping \(f\) is closed, then there is a neighborhood \(Uy\) such that \(f^{-1}(Uy)\subseteq Y\setminus \overline V^{\beta}\), since \(f^{-1}y\subseteq X\setminus \overline V^{\beta}\). By (1), similarly to the preceding, we obtain that \(y\in Uy\cap \overline{f(V\cap X)}\). Therefore the set \(H'=Uy\cap f(V\cap X)\) is nonempty, and for any point \(y'\) of \(H'\) we again obtain a contradiction: \(f^{-1}y'\subseteq f^{-1}(Uy)\subseteq Y\setminus \overline V^{\beta}\) and \(f^{-1}(y')\cap V\ne\varnothing\). Thus, if \(y\in Y\), then the preimage \(f_\beta^{-1}y\) is connected.

Second case: \(y\in \beta Y\setminus Y\). By (2), the set
\(H''=Oy\cap \overline{f(U\cap X)\cap \overline{f(V\cap X)}}\) is nonempty. Let \(y'\in H''\). We now show that \(f_\beta^{-1}(y')\cap \overline U^{\beta}\ne\varnothing\). Assuming the contrary, by closedness of the mapping \(f_\beta\), we find a neighborhood \(Oy'\) such that \(f_\beta^{-1}(Oy')\cap \overline U^{\beta}=\varnothing\). Then the set \(H'''=Oy\cap Oy'\cap \overline{f(U\cap X)}\) would be nonempty, and for any point \(y''\) of \(H'''\) we would have \(f_\beta^{-1}y''\subseteq f_\beta^{-1}(Oy')\subseteq \beta X\setminus \overline U^{\beta}\) and \(f_\beta^{-1}(y'')\cap U\ne\varnothing\), which is impossible. Thus, \(f_\beta^{-1}(y')\cap \overline U^{\beta}\ne\varnothing\). Similarly, \(f_\beta^{-1}(y')\cap \overline V^{\beta}\ne\varnothing\). At the same time, by (3), \(f_\beta^{-1}y'\subseteq f_\beta^{-1}(Oy)\subseteq \overline U^{\beta}\cup \overline V^{\beta}\). Since \(y'\in H''\subseteq Y\), the preimage \(f_\beta^{-1}y'\) is connected, which, in view of the three relations just proved, contradicts the condition \(\overline U^{\beta}\cap \overline V^{\beta}=\varnothing\), which the sets \(U\) and \(V\) satisfy. Thus, the supposition that the preimage of some point \(y\in\beta Y\) is disconnected leads, in both possible cases, to a contradiction. The theorem is proved.

Example 1. Let \(N\) be a countable space consisting of isolated points, let \(bN\) be any countable extension of it (for example, the extension \(N\cup \xi\) of P. S. Aleksandrov), let \(f\) be the identity mapping of the space \(N\) onto itself, and let \(f_\beta\) be its natural extension mapping \(\beta N\) onto \(bN\). Since the extension \(\beta N\) is uncountable, there exist points \(y\in bN\) with infinite preimages. At all such points the mapping \(f_\beta\) is not monotone, since \(\dim \beta N=0\), and hence \(\dim f_\beta^{-1}y=0\) for all points \(y\).

* \(\overline M^{\beta}\) denotes closure in the Čech extension, \(\overline M\) denotes closure in the space being extended, and \(\varnothing\) denotes the empty set.

Example 2. Let \(X=N\), and let \(Y\) be an arbitrary countable compactum, and let \(f\) be some one-to-one mapping of the space \(X\) onto \(Y\). It is discontinuous, not open, and not closed. The extended mapping \(f_\beta\) of the extension \(\beta X\) onto \(\beta Y=Y\) is nonmonotone by the same considerations as in Example 1.

Mathematical Institute
of the Siberian Branch of the Academy of Sciences of the USSR

Received
11 V 1960

References

1. P. S. Aleksandrov, UMN, 2, no. 1 (17) (1946).