Proximate fixed point property and operations (Q260525)

A compactuum is a compact metric space. Given a compactum \(X\) with metric \(d\), and \(\delta>0\), a continuous function \(f:X \rightarrow X\) is \(\delta\)-continuous if there exists \(\eta>0\) such that if \(d(u,x)<\eta\), then \(d(u,x)<\delta\).NEWLINENEWLINEThe compactum \(X\) has the proximate fixed point property (pfpp) if for every \(\varepsilon>0\), there exists a number \(\delta>0\) such that if \(f:X \rightarrow X\) is \(\delta\)-continuous, then there exists a point \(x \in X\) such that \(d(x,f(x))< \varepsilon\).NEWLINENEWLINEThe definition of pfpp was introduced by \textit{V. Klee} [Colloq. Math. 8, 43--46 (1961; Zbl 0101.15101)] who proved that if a compactum has the pfpp, then \(X\) has the fixed point property. The Warsaw circle is an example of a compactum with the fixed point property and without the pfpp.NEWLINENEWLINEIn the paper under review the author gives conditions under which products, cones, suspensions and joins of compacta have the pfpp. As an application, he shows that having span \(0\) is a sufficient conditon for all these spaces to have the pfpp.