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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