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Extending monotone mappings. (English) Zbl 0904.54009

Consider the following situation: one has a continuous map \(f:X\to Y\), where \(X\) and \(Y\) are dense subsets of \(C\) and \(D\) respectively. The general question is: what properties does a possible extension \(\widetilde f:C\to D\) of \(f\) share with \(f\) itself? As all spaces under consideration are completely regular, the extension \(\widetilde f\), if it exists, will be unique; one may therefore also consider \(\widetilde f\) given with \(\widetilde f[X]\subseteq Y\) and study the relationship between \(f=\widetilde f\restriction X\) and \(\widetilde f\). The authors investigate when ‘\(f\) is monotone’ implies ‘\(\widetilde f\) is monotone’ in the case when \(f\) is a surjection and, in addition, closed.
There are answers to several variations on the question: the first says that ‘\(\widetilde f\) is always monotone’ depends largely on the position of \(Y\) in \(D\). A nice internal characterization is: for any two disjoint closed sets \(A\) and \(B\) in \(Y\) the intersection \(\text{ cl}_D A\cap\text{ cl}_D B\) is disjoint from \(\text{ int}_D\text{ cl}_D (A\cup B)\). In case \(D\) is compact this is equivalent to \(D\) being a perfect compactification of \(Y\).
The second variation drops \(C\) and asks when one can find an extension \(C\) of \(X\) so that \(f\) has a monotone and perfect extension to \(C\); this is possible if there is an open subset \(O\) between \(Y\) and \(D\) that is an extension as in the previous paragraph.
The final result says that a monotone map \(f:X\to Y\) can always be extended to a monotone map between compactifications whose weight is at most the maximum of the weights of \(X\) and \(Y\).
Reviewer: K.P.Hart (Delft)

MSC:

54C20 Extension of maps
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
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