Metric Spaces

Metric spaces are the working environment for virtually all of modern analysis. In functional analysis, spaces of functions (continuous functions with the sup-metric, square-integrable functions with an $L^2$-metric) are metric spaces whose completeness — see Completeness and Cauchy Sequences — underlies the existence theory for differential equations via fixed-point arguments. In geometry, Riemannian manifolds carry a metric induced by their curvature, and the study of "metric geometry" (Gromov-Hausdorff distances between spaces themselves) extends the idea to comparing entire spaces rather than points. In computer science, metric spaces model similarity search: nearest-neighbor algorithms, clustering, and recommendation systems all depend on choosing an appropriate distance function on data (Euclidean distance on vectors, edit distance on strings, cosine distance on embeddings). In statistics and machine learning, metrics on probability distributions (Wasserstein distance, total variation distance) quantify how different two distributions are, feeding directly into generative modeling and hypothesis testing. In cryptography and coding theory, the Hamming metric on binary strings measures how many bits two codewords differ in, driving error-correction design. Physics uses metric structures both literally (spacetime intervals in relativity, though there the "distance" can be negative, a genuinely different structure called a pseudo-Riemannian metric) and analogically (phase-space distances in dynamical systems). All of this rests on the same four axioms introduced here, and all of it feeds forward into the topological machinery of Open and Closed Sets in Metric Spaces and eventually the fully abstract setting of Topological Spaces and Open Sets.

Ask the tutor: Can you show me exactly how open balls in a metric space generate a topology, and why not every topology can arise this way?

Think of the four axioms as encoding exactly what you would demand of any sensible "distance" before trusting it. Non-negativity says distances are never negative — you cannot be a negative amount of effort away from somewhere. Identity of indiscernibles says the only point at distance zero from $x$ is $x$ itself; distance is a genuine separator of points, not a fuzzy approximate one. Symmetry says the distance from $x$ to $y$ equals the distance from $y$ to $x$ — a strong assumption in some real-world settings (a one-way street violates this!), but built into the mathematical notion of metric. The triangle inequality is the deepest axiom: it says that going directly from $x$ to $z$ is never longer than detouring through $y$. Geometrically, in the plane, this is just the fact that any side of a triangle is at most the sum of the other two. Once you accept these four rules, an entire geometry crystallizes around the set $X$. The open balls $B(x,r)$ play the role that "intervals around a point" play on the real line: they are the neighborhoods that will be used to define limits, continuity, and open sets. Crucially, the shape of a ball depends entirely on the metric, not just on the underlying set. In $\mathbb{R}^2$ with the usual Euclidean distance, a ball is a round disk; with a different but equally legitimate metric, the "ball" of the same radius can be a diamond or a square. The figure below shows three different unit balls around the origin in $\mathbb{R}^2$, all valid metric-space geometries living on the exact same underlying set of points.

Example 1: The sup-metric on bounded functions, and a genuine triangle-inequality check with unequal terms. Let $X = B[0,1]$ be the set of all bounded real-valued functions on $[0,1]$. Define

This is finite since $f,g$ are bounded (so $|f(t)-g(t)| \le |f(t)| + |g(t)| \le M_f + M_g$ for all $t$, giving a finite upper bound, hence a finite supremum). Let's verify all four axioms and then do a genuinely asymmetric numerical check. Non-negativity: $|f(t)-g(t)| \ge 0$ for every $t$, so the sup of these values is $\ge 0$. Identity of indiscernibles: if $f=g$ then $|f(t)-g(t)|=0$ for all $t$, so the sup is $0$. Conversely if $d(f,g)=0$ then $\sup_t |f(t)-g(t)| = 0$ forces $|f(t)-g(t)| = 0$ for every $t$ (if it were positive at some $t_0$, the sup would be at least that positive value), so $f(t) = g(t)$ for all $t$, i.e. $f = g$. Symmetry: $|f(t)-g(t)| = |g(t)-f(t)|$ pointwise, so the sups agree. Triangle inequality: for every $t \in [0,1]$,

using the ordinary triangle inequality for real numbers pointwise, then bounding each term by its supremum. The right side does not depend on $t$, so it is an upper bound for the set $\{|f(t)-h(t)| : t \in [0,1]\}$, hence

Now a concrete, deliberately asymmetric numerical check. Let

Compute $d(f,g) = \sup_{t\in[0,1]} |t^2 - 3t + 1|$. The function $\phi(t) = t^2-3t+1$ has derivative $2t - 3 < 0$ on $[0,1]$, so $\phi$ is strictly decreasing; its extreme values on $[0,1]$ occur at the endpoints: $\phi(0) = 1$, $\phi(1) = 1 - 3 + 1 = -1$. So $|\phi(t)|$ is maximized at an endpoint, giving $d(f,g) = \max(|1|,|-1|) = 1$. Compute $d(g,h) = \sup_t |3t - 1 - \sin(\pi t)|$. At $t=0$: $|{-1} - 0| = 1$. At $t=1$: $|2 - 0| = 2$. At $t = 1/2$: $|0.5 - 1| = 0.5$. Since $\sin(\pi t) \in [0,1]$ throughout $[0,1]$ and $3t-1$ ranges over $[-1,2]$, the maximum gap is at $t=1$, giving $d(g,h) = 2$ (checking nearby points confirms the maximum of the increasing-ish expression $3t-1-\sin(\pi t)$ occurs at $t=1$, since its derivative $3 - \pi\cos(\pi t)$ is positive whenever $\cos(\pi t) < 3/\pi \approx 0.955$, true for all $t$ close enough to $1$, and one checks the derivative is positive throughout $(0,1)$ except possibly very near $t=0$ where it's still $3-\pi>0$; so the expression is increasing and its max is at $t=1$). Compute $d(f,h) = \sup_t |t^2 - \sin(\pi t)|$. At $t = 1$: $|1 - 0| = 1$. At $t=0$: $|0-0|=0$. At $t=0.8$: $t^2 = 0.64$, $\sin(0.8\pi) = \sin(144^\circ) \approx 0.5878$, difference $\approx 0.0522$. The maximum magnitude on $[0,1]$ is at $t=1$, giving $d(f,h) = 1$. The triangle inequality claims $d(f,h) \le d(f,g) + d(g,h)$, i.e.

which indeed holds — with plenty of room to spare, illustrating that the inequality need not be tight.

Example 2: The discrete metric and a non-Euclidean ball structure. Let $X$ be any nonempty set (say $X = \{a,b,c,d\}$, four abstract symbols with no numerical structure at all), and define

Check the axioms: non-negativity and symmetry are immediate from the definition, and identity of indiscernibles holds by construction ($d(x,y)=0$ exactly when $x=y$). For the triangle inequality, take any $x,y,z$. If $x = z$, then $d(x,z) = 0 \le d(x,y)+d(y,z)$ automatically since the right side is a sum of non-negative numbers. If $x \ne z$, then $d(x,z) = 1$; we must show $d(x,y) + d(y,z) \ge 1$. If $y = x$ then since $y = x \ne z$ we get $d(y,z) = d(x,z) = 1$, so the sum is at least $1$. If $y \ne x$, then $d(x,y) = 1$ already, so the sum is at least $1$. Either way $d(x,y)+d(y,z)\ge 1 = d(x,z)$, so the triangle inequality holds in every case. Now compute balls. For any $x \in X$ and $0 < r \le 1$, $B(x,r) = \{x\}$, since the only point at distance $<r\le 1$ from $x$ is $x$ itself (any other point is at distance exactly $1 \ge r$). For $r > 1$, $B(x,r) = X$, since every point (including $x$) is at distance $<r$. So in the discrete metric every ball is either a single point or the entire space — there is no "shrinking gradually," unlike Euclidean balls, which illustrates how differently the same four axioms can behave on different spaces.