# Metric Space

How far apart are two [functions](/page/Function)? Two matrices? Two probability [distributions](/page/Distribution)? On the real line, the distance between numbers $x$ and $y$ is $|x - y|$, and the entire edifice of real analysis --- convergence, [continuity](/page/Continuity), compactness --- rests on this single measurement. But the real line is only one setting among many. In the space $C[0,1]$ of continuous functions on the unit interval, the "distance" between two functions $f$ and $g$ might be the largest vertical gap $\sup_{t \in [0,1]} |f(t) - g(t)|$, or the total area between their graphs $\int_0^1 |f(t) - g(t)| \, dt$. In a network, distance might count hops rather than metres. The concept of a metric space isolates the three properties that make any notion of distance analytically useful --- positivity, symmetry, and the triangle inequality --- and demonstrates that once these three axioms hold, the full machinery of convergence, continuity, compactness, and completeness follows without reference to coordinates, dimensions, or algebraic structure.

[motivation] ### Beyond Euclidean distance In a first analysis course, every distance is $|x - y|$ on $\mathbb{R}$ or $\|x - y\|$ on $\mathbb{R}^n$. This is sufficient for studying [sequences](/page/Sequence) of numbers and functions of a real variable, but it becomes limiting as soon as the objects of study are themselves more complex. Consider the space of all continuous functions $f: [0,1] \to \mathbb{R}$. Two functions $f$ and $g$ might agree everywhere except on a tiny interval near $t = 1/2$, where $f$ spikes sharply above $g$. Are they "close"? That depends on how we measure: the supremum distance $\sup |f(t) - g(t)|$ says they are far apart (the spike is tall), while the $L^1$ distance $\int_0^1 |f(t) - g(t)| \, dt$ says they are close (the spike is narrow, so the area is small). Different notions of distance on the same underlying set can lead to different convergent sequences, different continuous functions, and different compact subsets. A framework that accommodates all such notions simultaneously is not a luxury --- it is a necessity. ### What makes a "distance" useful? Not every assignment of non-negative numbers to pairs of points deserves to be called a distance. Three requirements emerge from examining what makes $|x - y|$ so effective on $\mathbb{R}$. First, **positivity and identity of indiscernibles**: $d(x, y) \geq 0$ always, and $d(x, y) = 0$ if and only if $x = y$. Without this, distinct points could be "distance zero apart," collapsing the ability to distinguish them. Relaxing this condition leads to pseudometrics, which arise naturally in quotient constructions but sacrifice the power to separate points. Second, **symmetry**: $d(x, y) = d(y, x)$. The distance from London to Paris is the same as from Paris to London. Dropping symmetry leads to quasimetrics, which model situations like travel times on one-way streets, but which require separate theories for "forward convergence" and "backward convergence." Third, the **triangle inequality**: $d(x, z) \leq d(x, y) + d(y, z)$. This is the axiom with the deepest consequences. It prevents "shortcuts" --- going from $x$ to $z$ via $y$ can never be cheaper than going directly. Every proof in metric space theory that involves approximation --- and that is nearly all of them --- relies on the triangle inequality to control accumulated errors. The $\varepsilon/3$ argument, the passage from pointwise to uniform convergence, the proof that Cauchy sequences are bounded: all depend on chaining triangle inequality estimates. ### The payoff Once a set $X$ is equipped with a function $d$ satisfying these three axioms, the entire toolkit of analysis becomes available. Open balls define a topology; convergence is characterised by $d(x_n, x) \to 0$; continuity means $\varepsilon$-$\delta$ with $d$ in place of absolute value; Cauchy sequences and completeness generalise directly. The remarkable fact is that these constructions do not depend on the specific formula for $d$ --- only on the three axioms. A single proof of the Heine-Cantor theorem, for instance, works simultaneously for real-valued functions on an interval, for maps between surfaces, and for operators between Banach spaces. The metric space framework is the engine that makes this kind of unification possible. [/motivation]

## Definition

[definition:Metric Space] A **metric space** is an ordered pair $(X, d)$, where $X$ is a non-empty set and $d: X \times X \to \mathbb{R}$ is a function, called the **metric** (or **distance function**), satisfying the following three axioms for all $x, y, z \in X$: 1. **Positivity and identity of indiscernibles.** \begin{align*} d(x, y) \geq 0, \quad \text{and} \quad d(x, y) = 0 \iff x = y. \end{align*} 2. **Symmetry.** \begin{align*} d(x, y) = d(y, x). \end{align*} 3. **Triangle inequality.** \begin{align*} d(x, z) \leq d(x, y) + d(y, z). \end{align*} The elements of $X$ are called **points**, and $d(x, y)$ is called the **distance** between $x$ and $y$. [/definition]

The three axioms are independent: no two of them imply the third, and each rules out a distinct pathology. Axiom 1 ensures that the metric separates points --- it functions as an identity test. Axiom 2 ensures that the relation "being close to" is symmetric. Axiom 3 ensures that distances compose in a controlled way, which is essential for every approximation argument in the subject.

A consequence of Axioms 1 and 3 is the **reverse triangle inequality**: for all $x, y, z \in X$,

\begin{align*} |d(x, z) - d(y, z)| \leq d(x, y). \end{align*}

To see this, apply the triangle inequality twice: $d(x, z) \leq d(x, y) + d(y, z)$ gives $d(x, z) - d(y, z) \leq d(x, y)$, and swapping the roles of $x$ and $y$ gives $d(y, z) - d(x, z) \leq d(y, x) = d(x, y)$. Taking the maximum of the two inequalities yields the result. The reverse triangle inequality shows that the distance function $d$ is Lipschitz continuous (with constant $1$) in each variable separately, a fact used repeatedly in the sequel.

[example:Euclidean Metric] The **Euclidean metric** on $\mathbb{R}^n$ is defined by \begin{align*} d_2: \mathbb{R}^n \times \mathbb{R}^n &\to \mathbb{R} \\ (x, y) &\mapsto \left( \sum_{i=1}^{n} (x_i - y_i)^2 \right)^{1/2}, \end{align*} where $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_n)$. Positivity and symmetry are immediate. The triangle inequality $d_2(x, z) \leq d_2(x, y) + d_2(y, z)$ is the Minkowski inequality for $p = 2$, which in turn follows from the Cauchy-Schwarz inequality. When $n = 1$, this reduces to the familiar $|x - y|$ on the real line. [/example]

[example:Supremum Metric On Bounded Functions] Let $S$ be a non-empty set and let $B(S)$ denote the set of all bounded functions $f: S \to \mathbb{R}$. Define \begin{align*} d_\infty: B(S) \times B(S) &\to \mathbb{R} \\ (f, g) &\mapsto \sup_{s \in S} |f(s) - g(s)|. \end{align*} The supremum exists because $f - g$ is bounded. Positivity and symmetry follow from those of $|\cdot|$. For the triangle inequality, let $f, g, h \in B(S)$ and $s \in S$. Then \begin{align*} |f(s) - h(s)| \leq |f(s) - g(s)| + |g(s) - h(s)| \leq d_\infty(f, g) + d_\infty(g, h). \end{align*} Taking the supremum over $s$ gives $d_\infty(f, h) \leq d_\infty(f, g) + d_\infty(g, h)$. This metric is fundamental in analysis: convergence in $d_\infty$ is precisely [uniform convergence](/page/Uniform%20Convergence), and the resulting metric space structure on $C[0,1]$ underpins the Arzela-Ascoli theorem and the theory of [Banach spaces](/page/Banach%20Space). [/example]

[example:Discrete Metric] On any non-empty set $X$, define \begin{align*} d_{\text{disc}}: X \times X &\to \mathbb{R} \\ (x, y) &\mapsto \begin{cases} 0 & \text{if } x = y, \\ 1 & \text{if } x \neq y. \end{cases} \end{align*} Axioms 1 and 2 are immediate. For the triangle inequality, if $x = z$ then $d_{\text{disc}}(x, z) = 0 \leq d_{\text{disc}}(x, y) + d_{\text{disc}}(y, z)$. If $x \neq z$, then $y$ differs from at least one of $x$ or $z$, so $d_{\text{disc}}(x, y) + d_{\text{disc}}(y, z) \geq 1 = d_{\text{disc}}(x, z)$. The discrete metric is the coarsest possible: it detects only whether points are equal or distinct, nothing about "how far apart" they are. Every subset of $(X, d_{\text{disc}})$ is open, so the induced topology is the discrete topology. A sequence converges in this metric if and only if it is eventually constant. [/example]

## Open Sets and Topology

The concept of an open set arises naturally from the metric: a set is open if every point in it has room to move in all directions without leaving the set. This is formalised through open balls.

[definition:Open Ball] Let $(X, d)$ be a metric space, let $x \in X$, and let $r > 0$. The **open ball** of radius $r$ centred at $x$ is the set \begin{align*} B_r(x) = \{ y \in X : d(x, y) < r \}. \end{align*} [/definition]

In $\mathbb{R}$ with the standard metric, $B_r(x) = (x - r, x + r)$ is an open interval. In $\mathbb{R}^2$ with the Euclidean metric, $B_r(x)$ is an open disc. In $\mathbb{R}^2$ with the taxicab metric $d_1(x, y) = |x_1 - y_1| + |x_2 - y_2|$, the "ball" $B_r(x)$ is a diamond (square rotated $45°$). The shape of open balls depends on the metric, not just on the underlying set.

[definition:Open Set In A Metric Space] Let $(X, d)$ be a metric space. A subset $U \subseteq X$ is **open** if for every $x \in U$, there exists $r > 0$ such that $B_r(x) \subseteq U$. [/definition]

Openness captures the idea that $U$ has no "[boundary](/page/Boundary) points" among its members: every point of $U$ is an interior point, surrounded by a small ball entirely contained in $U$.

The collection of all [open sets](/page/Open%20Set) in a metric space $(X, d)$ satisfies three properties that define a topology:

(i) The empty set $\varnothing$ and the full space $X$ are open.