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- $d(x,y) \geq 0$ for all $x,y \in X$.
- $d(x,y) = 0 \Leftrightarrow x = y$.
- $d(x,y) = d(y,x)$.
- (The triangle inequality): If $x,y,z \in X$, then:
$$d(x,z) \leq d(x,y) + d(y,z).$$
In a metric space $(X,d)$, the nonnegative real number $d(x,y)$ is called the distance from $x$ to $y$ (or between $x$ and $y$).
To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Metric geometry