Topological properties of asymptotically stable sets (Q984078)

The object of the paper is the continuous time invariant semi-flow \(\varphi:R_{\geq 0}\times M\mapsto M\) on a locally compact compact topological space \(M\). First the following notions are reminded or introduced: asymptotic stability of a set and its attraction domain which is open and invariant, retract, deformation retract, weak deformation retract, neighborhood retract, neighborhood deformation retract, strong neighborhood deformation retract. Then the following results are proved. Theorem. Let \(X\subseteq M\) be a weak deformation retract of \(M\) of \(Y\subseteq M\). Then: a) if \(X\) is a neighborhood retract of \(M\), \(X\) is a retract of \(Y\); b) if \(X\) is a neighborhood deformation retract of \(M\), then \(X\) is a deformation retract of \(Y\); c) if \(X\) is a strong neighborhood retract of \(M\), then \(X\) is a strong deformation retract of \(Y\). Theorem. Suppose \(K\) is a compact asymptotically set for the dynamical system \((M,\varphi)\) with domain of attraction \(A\). Then \(K\) is a weak deformation retract of \(A\).