Some fixed point theorems for nonconvex spaces (Q1380967)

Let \(E\) be a topological vector space over the scalar field \(\mathbb{R}\) or \(\mathbb{C}\). Let \(\tau\) and \(\tau_w\) denote the original and weak topologies of \(E\). For a nonempty subset \(X\subset E\), \((X,\tau_w)\) (resp. \((X,\tau)\)) denotes \(X\) with the relative topology \(\tau_w\) (resp. \(\tau\)). The main result is the following proposition. Proposition 2.1. Let \(X\) be a nonempty compact (in topology \(\tau\)) subset of \(E\). If \((X,\tau)\) is Hausdorff, then \((X,\tau_w)= (X,\tau)\). The authors derive fixed point theorems of Ky Fan, Kim, Kaczyński, Kelly, Namioka from this proposition and prove a fixed point theorem for multivalued functions.