On the \(\delta\)-continuous fixed point property (Q756744)

The authors study retractions and the fixed point property for \(\delta\)- continuous functions, which they defined independently in earlier papers. A function \(f:X\to Y\) if \(\delta\)-continuous if for each \(x\in X\) and neighborhood V of \(f(x)\) there is a neighbourhood U of x such that \(f[Int(\bar U)]\subset Int(\bar V)\). This is one of a hierarchy of definitions: \(\delta\)-continuous \(\Rightarrow\) almost continuous \(\{f(U)\subset Int(\bar V)\}\Rightarrow\vartheta\)-continuous \(\{f(\bar U)\subset \bar V\}\Rightarrow\) weakly continuous \(\{f(U)\subset \bar V\}\). The authors give short proofs of nine propositions and theorems on retractions and the fixed point property for \(\delta\)-continuous functions. These notions are defined in the usual way. They use arguments from general rather than algebraic topology.