Metric Spaces

A metric space is a set equipped with a notion of distance satisfying natural axioms. Metric spaces generalize $\mathbb{R}^n$ with the Euclidean distance and provide the abstract framework for analysis. Concepts like convergence, continuity, and compactness extend seamlessly to metric spaces, unifying diverse areas of mathematics.

Definition

Definition8.1Metric space

A metric space is a pair $(X, d)$ where $X$ is a set and $d : X \times X \to [0, \infty)$ is a metric (or distance function) satisfying:

Positivity: $d(x, y) \geq 0$ , with equality iff $x = y$ .
Symmetry: $d(x, y) = d(y, x)$ .
Triangle inequality: $d(x, z) \leq d(x, y) + d(y, z)$ .

ExampleEuclidean metric on R^n

$d(x, y) = \|x - y\| = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}$ is the standard Euclidean metric on $\mathbb{R}^n$ .

ExampleDiscrete metric

On any set $X$ , define $d(x, y) = 0$ if $x = y$ and $d(x, y) = 1$ if $x \neq y$ . This is the discrete metric — every subset is open!

ExampleMetric on C([a, b])

On $C([a, b])$ (continuous functions on $[a, b]$ ), define $d(f, g) = \sup_{x \in [a, b]} |f(x) - g(x)|$ (the uniform norm). This makes $C([a, b])$ a metric space.

ExampleL¹ metric on sequences

On the space of absolutely summable sequences, $d(x, y) = \sum_{n=1}^\infty |x_n - y_n|$ defines a metric (the $\ell^1$ metric).

Convergence and continuity

Definition8.2Convergence in metric spaces

A sequence $(x_n)$ in $(X, d)$ converges to $x \in X$ if $d(x_n, x) \to 0$ as $n \to \infty$ .

Definition8.3Continuity

A function $f : (X, d_X) \to (Y, d_Y)$ is continuous at $x \in X$ if for every $\epsilon > 0$ , there exists $\delta > 0$ such that $d_X(x, y) < \delta$ implies $d_Y(f(x), f(y)) < \epsilon$ .

RemarkGeneralization

All definitions from $\mathbb{R}$ (open sets, closed sets, compactness, completeness) extend naturally to metric spaces by replacing $|x - y|$ with $d(x, y)$ .

Open and closed sets

Definition8.4Open ball

The open ball of radius $r$ centered at $x$ is $B_r(x) = \{y \in X \mid d(x, y) < r\}$ .

Definition8.5Open set

$U \subseteq X$ is open if for every $x \in U$ , there exists $r > 0$ such that $B_r(x) \subseteq U$ .

ExampleAll sets are open in discrete metric

In the discrete metric, every set is open (since $B_{1/2}(x) = \{x\}$ ).

Summary

Metric spaces provide the abstract framework for analysis:

Distance function $d$ satisfying positivity, symmetry, triangle inequality.
Convergence, continuity, open/closed sets defined via $d$ .
Examples: $\mathbb{R}^n$ , function spaces, sequence spaces.
Unifies diverse mathematical structures.

See Complete Metric Spaces and Compactness.