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Metric Space

$\newcommand{R}{\mathbb{R}}$ A metric space is a space where the structure comes from a function $d$ which is called a metric.

The ordered pair $(M, d)$ is a metric space where $M$ is a set and $d$ is $d: M \times M \rightarrow \R$ and $d$ satisfies the following

  1. $d(x, x) = 0$
  2. Positive: $d(x, y) > 0 : x \ne y$
  3. Symmetric: $d(x, y) = d(y, x)$
  4. Triangle Inequality: $d(x, z) \le d(x, y) + x(y, z)$

Relation to topology

Every metric space is also a topological space

That is given a metric $d$ for some set $X$, we can always define a topology $\tau$.

We call this topology the metric topology

A metric is a very easy way to define bases of a topology.

$B_r(y) = {x \in X : d(x, y) < r}, r> 0, y \in X$

The topology is the union of these open balls