Definition

Consider a complete Metric Space $(X, T, d)$ and a Contraction Map on it $f : X \to X$ . Then there exists a unique point $x \in X$ such that $f (x) = x$ . $\exists! x \in X s.t. f (x) = x$ where the point $x$ is called the fixed point of $f$ .

For a Metric Space $(X, d)$ there exists complete Metric Space $(Y, d^{'})$ and an isometry $e : X \to Y$ such that $e (X)$ is a dense subset of $Y$ .