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Extension of maps and ordering. (English) Zbl 0952.54010
Two continuous mappings \(\varphi,\psi: X\to T\) are equivalent, \(\varphi\simeq \psi\), if there is a homeomorphism \(\pi: \varphi(X)\to \psi(X)\) between the images such that \(\pi \varphi= \psi\).
Let \(Y\) be a closed subset of \(X\), \(\psi:Y\to \mathbb{R}^\omega\) and \(\varphi: X\to \mathbb{R}^\omega\) continuous mappings (\(\mathbb{R}^\omega\) the countable infinite product of copies of the real numbers \(\mathbb{R}\)). The following type of extension of the map \(\psi\) is treated.
When is there an extension \(\eta: X\to \mathbb{R}^\omega\) of \(\psi\) such that \(\eta\mid X\smallsetminus Y\simeq \varphi\mid X\smallsetminus Y\)? The answer is yes if
(i) there is a map \(\pi: \varphi(X)\to \psi(X)\) such that \(\pi\varphi= \psi\);
(ii) \(\varphi(Y)\) and \(\psi(Y)\) are closed in \(\mathbb{R}^\omega\);
(iii) \(Y= \varphi^{-1} (\varphi(Y))\).
Another theorem treats an analogous question when \(X\) is a real algebraic variety, \(Y\) a closed algebraic subvariety and \(\varphi\) and \(\psi\) are algebraic maps.
MSC:
54C20 Extension of maps
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