The SlugMath Wiki is under heavy development!

# Def/Metric space

Definition of Metric space: A metric space is an ordered pair $(X, d)$, where $X$ is a set, and $d \colon X \times X \rightarrow \RR$ is a metric, that is:

• $d(x,y) \geq 0$ 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 $y$).

## Logical Connections

This definition logically relies on the following definitions and statements: Def/Real number, Def/Finite (set), State/Truth by false premise

The following statements and definitions logically rely on the material of this page: Def/Interior, and Def/Limit (sequence)

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