  In this paper, we introduce the notion of topologically Banach contraction
mapping defined on an arbitrary topological space X with the help of a
continuous function $g:X\times X\rightarrow \mathbb{R}$ and investigate the
existence of fixed points of such mapping. Moreover, we introduce two types of
mappings defined on a non-empty subset of X and produce sufficient conditions
which will ensure the existence of best proximity points for these mappings.
Our best proximity point results also extend some existing results from metric
spaces or Banach spaces to topological spaces. More precisely, our newly
introduced mappings are more general than that of the corresponding notions
introduced by Bunlue and Suantai [Arch. Math. (Brno), 54(2018), 165-176]. We
present several examples to validate our results and justify its motivation. To
study best proximity point results, we introduce the notions of g-closed,
g-sequentially compact subsets of X and produce examples to show that there
exists a non-empty subset of X which is not closed, sequentially compact under
usual topology but is g-closed and g-sequentially compact.
