Continuous \(\varepsilon\)-selection (Q2812517)

A nonempty subset \(M\) of a seminormed linear space \((X,\|\cdot\|)\) is called infinitely connected if for any \(n\in\mathbb N\), any continuous mapping \(\varphi:\partial B\to M\) has a continuous extension \(\tilde \varphi:B\to M\), where \(B\) denotes the closed unit ball in \(\mathbb R^n\) and \(\partial B\) its boundary. A set \(M\) is called \({B^\circ}\)-infinitely connected if its intersection with any open ball is either empty or infinitely connected. Let \(\varepsilon>0\). A mapping \(\varphi:X\to M\) is called an additive (multiplicative) \(\varepsilon\)-selection if, for any \(x\in X,\; \|x-\varphi(x)\|\leq\varepsilon+\rho(x,M)\) (resp. \(\|x-\varphi(x)\|\leq(1+\varepsilon)\cdot\rho(x,M)\)), where \(\rho(x,M)=\inf\{\|x-y\|:y\in M\}\). The author gives necessary and sufficient conditions for the existence of continuous additive and multiplicative \( \varepsilon\)-selections on closed sets. For example, if \(X\) is a complete seminormed space, then the following conditions are equivalent: {\parindent=0.7cm\begin{itemize}\item[(a)] the set \(M\) is \({B^\circ}\)-infinitely connected; \item[(b)] for any lower semi-continuous function \(\psi:X\to\overline{\mathbb R}\) such that \(\psi(x)>\rho(x,M)\) for all \(x\in X\), there exists \(\varphi\in C(X,M)\) such that \(\|\varphi(x)-x\|<\psi(x)\) for all \(x\in X\); \item[(c)] for any \(\varepsilon>0\) there exists \(\varphi\in C(X,M)\) such that \(\|\varphi(x)-x\|<\varepsilon+\rho(x,M)\) for all \(x\in X\) (a continuous additive \(\varepsilon\)-selection); \item[(d)] for any lower semi-continuous function \(\theta:X\to\overline{\mathbb R}_+\) such that there exists \(\varphi\in C(X,M)\) such that \(\|\varphi(x)-x\|\leq \theta(x)\cdot\rho(x,M)\), for all \(x\in X\); \item[(e)] for any \(\varepsilon>0\) there exists \(\varphi\in C(X,M)\) such that \(\|\varphi(x)-x\|<(1+\varepsilon)\cdot\rho(x,M)\) for all \(x\in X\) (a continuous multiplicative \(\varepsilon\)-selection). NEWLINENEWLINE\end{itemize}} As consequences, one obtains the existence of continuous selections for two classes of set-valued mappings \(F:X\rightrightarrows Y\), called lower stable and stable, where \(X\), \(Y\) are seminormed linear spaces, with \(Y\) complete for stable \(F\), and finite dimensional for \(F\) locally stable.