The properties of the multi-valued domination of metric spaces
In this paper the author studies properties of multi-valued domination of metric spaces and, in particular, examines relationships and similarities between absolute multi-retracts (AMR) (absolute neighborhood multi-retracts (ANME)) and absolute retracts (AR) (absolute neighborhood retracts (ANR)). The main theorem states: for every compact space \(X\) and \(Y\in \text{ANR}\) (\(Y\in \text{AR}\)), if \(Y \geq\!\!\circ_{tsa} X\) (i.e., \(Y\) strongly multi-dominates to the right over \(X\) by a mapping of TSA-type), then \(X\in \text{ANR}\) (\(X\in \text{AR}\)). Here condition \(Y \geq\!\!\circ_{tsa} X\) means there exist an admissible map of TSA-type \(\varphi: X\multimap Y\) (a map admitting a selected pair \((p, q) \subset \varphi\) and a metric space \(Z\) such that \(p: Z\to X\) is a cell-like map) and a continuous map \(r: Y\to X\) such that \(r(\varphi(x)) = \{x\}\) for each \(x\in X\). Using an author's earlier result that \(X\in\text{ANMR}\) (\(X\in\text{AMR}\)) is equivalent to the condition \(U \geq\!\!\circ_{tsa} X\) (\(E_1 \geq\!\!\circ_{tsa} X\)), where \(E\) and \(E_1\) are locally convex spaces and \(U\subset E\) is an open set, the main theorem implies that if \(X\) is a finitely compact or countably dimensional compact metric space and \(X\in \text{ANMR}\) (\(X\in\text{AMR}\)), then \(X\in\text{ANR}\) (\(X\in\text{AR}\)). Furthermore, introducing the notions of homotopy of TSA-type mapping, TSA-contractibility, and locally TSA-contractibility, the authors prove the following two theorems: (i) for every metric space \(X\) and contractible metric space \(Y\), if \(Y \geq\!\!\circ_{tsa} X\), then \(X\) is TSA-contractible; (ii) for every metric space \(X\) and ANR \(Y\), if \(Y \geq\!\!\circ_{tsa}^m X\), which means \(Y \geq\!\!\circ_{tsa} X\) and for each \(x\in X\), \(\varphi(x)\) is compact and movable, then \(X\) is locally TSA-contractible.
scientific article; zbMATH DE number 6150728