Metric Space

Definition

A metric space is an ordered pair $(X, d)$ where $X$ is a set and $d: X \times X \to \mathbb{R}$ is a metric on $X$.

Examples

• $(\mathbb{R}^n, d)$ with the Euclidean metric $d(x,y) = \sqrt{\sum (x_i - y_i)^2}$
• Any normed vector space $(V, \|\cdot\|)$ with $d(x,y) = \|x - y\|$
• The discrete metric space: $d(x,y) = \begin{cases} 0 & x = y \\ 1 & x \neq y \end{cases}$

Key Concepts

• A subset $U \subseteq X$ is open if for every $x \in U$, there exists $\varepsilon > 0$ such that $B(x, \varepsilon) \subseteq U$
• The open ball of radius $\varepsilon$ centered at $x$ is $B(x, \varepsilon) = \{y \in X : d(x, y) < \varepsilon\}$
• Every metric space is a topological space via the metric topology

Related Concepts

• Metric — the distance function on a metric space
• Norm — every norm induces a metric
• Topological Space — generalization of metric spaces