Metric space

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A metric space is an ordered pair $(X,d)$ where $X$ is some set and $d$ is a function $d \colon X \times X \rightarrow [0,\infty)$, called a metric, with the following properties:

  1. For all $x,y \in X$, $d(x,y)=d(y,x)$.
  2. For all $x,y \in X$, $d(x,y)=0$ if and only if $x=y$.
  3. For all $x,y,z \in X$, $d(x,y) \leq d(x,z)+d(z,y)$.

Metric spaces can be equipped with a natural topology whose basic open sets are the balls $B_{\epsilon}(x)$, where $B_{\epsilon}(x) = \{ y \in X \colon d(x,y)<\epsilon\}$.