Best proximity point theorems in topological spaces

Abstract

Let A, B be nonempty subsets of a metric space X and $$f:A\rightarrow B$$ be a mapping. A point $$x_0\in A$$ is a best proximity point of f if $$d(x,f(x))=\mathrm{dist}(A,B):=\{d(a,b):a\in A,b\in B\}$$. It is worth mentioning that the metric function d plays a vital role in defining the notion of best proximity points. In this manuscript, we introduced a notion of best proximity points in arbitrary topological spaces and established few best proximity point theorems. Our main result generalizes the well-known Edelstein’s fixed point theorem for contractive mappings.