Metric Space

A pair $(X, \rho)$, where $\rho$ is a metric considered metric space.

Connection with topology

• Each metric space is a topological-space with topology built by topology-base ${O_{\varepsilon}(x): \forall \varepsilon,, \forall x}$, where $O_{\varepsilon}(x)$ is an open-ball.

• Then the set $G$ is an open-set $\iff \forall x \in S: \exists R \hookrightarrow O_R(x) \subset G$.

• Each metric space satisfies the first axiom-of-countability with ${O_{\frac{1}{n}}(x)}$ as a definitive family of point-neighborhood.

• Metric space is separable-space if exists a countable subspace $S \subset X$ such that $\forall \varepsilon,,\forall x \in X: \exists s \in S \hookrightarrow \rho(x, s) < \varepsilon$.

  See also metric-space-non-separability-criteria