A Banach Fixed Point Theorem for topological spaces

The purpose of this paper is to prove a fixed point theorem for Hausdorff first countable topological spaces and to show, that it is really a generalization of the following well-known Banach Fixed Point Theorem: Let f:(X, d)  ͢   (X, d) be a contraction mapping from a complete metric space into itself. Let x ∊ X. Then the sequence {f ⁽ⁿ⁾ (x)} converges to a point z₀ ∊ X and f(z₀) = z₀. Furthermore, z₀ is the unique fixed point for f. (The definitions of notions not defined here can be found in [1]).