Continuous \(\varepsilon\)-selection. (English) Zbl 1347.41047
Sb. Math. 207, No. 2, 267-285 (2016); translation from Mat. Sb. 207, No. 2, 123-142 (2016).
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:
- (a)
- the set \(M\) is \({B^\circ}\)-infinitely connected;
- (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 \(x\in X\);
- (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);
- (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 \(x\in X\);
- (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).
Reviewer: Costică Mustăţa (Cluj-Napoca)
MSC:
41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
54C65 | Selections in general topology |
28B20 | Set-valued set functions and measures; integration of set-valued functions; measurable selections |
54C60 | Set-valued maps in general topology |