13 Metric Spaces

Vector Spaces was explicit about what it left out: addition and scaling, with nothing in the axioms about how “long” a vector is or how “close” two vectors are. That’s not an oversight — it’s a different structure’s job. This chapter names it.

13.1 The problem, before the definition

Distance feels primitive, but it’s worth noticing how many different things “distance” ends up meaning even before leaving \(\mathbb{R}^n\). The ordinary (Euclidean) distance between two points is \(\sqrt{\sum_i (x_i-y_i)^2}\). A taxi driver on a city grid cares about a different quantity: \(\sum_i |x_i-y_i|\), the taxicab distance — you can’t cut diagonally through buildings. Both are legitimate, both apply to the same set \(\mathbb{R}^n\), and they disagree about which pairs of points count as “close.” Distance, it turns out, isn’t a fact about a set — it’s an extra structure you choose to put on one.

And it doesn’t require numbers to be involved at all. On any set \(X\), define \(d(x,y) = 0\) if \(x=y\) and \(d(x,y)=1\) otherwise — the discrete metric. It’s a perfectly valid notion of distance on a set of colors, a set of people, a set of anything, with no arithmetic in sight. Whatever “distance” turns out to mean formally, it has to be general enough to cover all three of these at once.

13.2 The abstraction

A metric space is a set \(X\) with a function \(d: X \times X \to \mathbb{R}_{\geq 0}\) (the metric) satisfying:

\(d(x,y) = 0\) if and only if \(x = y\).
Symmetry: \(d(x,y) = d(y,x)\).
Triangle inequality: \(d(x,z) \leq d(x,y) + d(y,z)\).

13.3 Why these axioms

The first says distinct points are genuinely distinguishable by distance — without it, “distance zero” could hold between two different points, undermining the entire idea that \(d\) measures separation. Symmetry says the distance from \(A\) to \(B\) is the distance from \(B\) to \(A\) — drop it and “distance” would depend on direction of travel, which contradicts everything the word suggests (it’s also the axiom the taxicab metric respects but a one-way road network wouldn’t, which is exactly why a one-way street network isn’t a metric space in this sense). The triangle inequality is the substantive one: it says a detour through a third point \(y\) is never shorter than going directly — \(d(x,z) \leq d(x,y)+d(y,z)\). Without it, “distance” would stop supporting the single most basic geometric intuition attached to the word: that a straight line (or its generalization) is the short way.

13.4 What follows: open balls, and a first bridge to what’s next

The open ball of radius \(r\) around a point \(x\), \(B(x,r) = \{y \in X : d(x,y) < r\}\), is what lets a metric space define “nearby” precisely: \(y\) is near \(x\) if it’s inside a small enough ball around \(x\). A set \(U \subseteq X\) is called open if every point of \(U\) has some open ball around it still entirely contained in \(U\) — no matter how close to \(U\)’s edge a point sits, there’s room to wiggle. This sounds like a technicality, but it’s the single most important consequence in this chapter: once “open set” is defined, a sequence \(x_n\) converges to \(x\) exactly when, for every open ball around \(x\), all but finitely many \(x_n\) eventually land inside it — convergence, stated without a single inequality, purely in terms of which sets are open. That restatement is not decorative: it’s exactly what Topological Spaces generalizes next, by asking whether “open set” can be taken as the primitive notion instead of distance at all.

13.5 If you’ve done real analysis

You already know convergence as \(d(x_n, x) \to 0\) — precisely the \(\varepsilon\)–\(\delta\)-flavored definition above, just phrased with a general \(d\) instead of \(|x-y|\) on \(\mathbb{R}\). Every fact you know about limits in \(\mathbb{R}\) that only used the distance \(|x-y|\), not anything special about real numbers, is secretly a fact about metric spaces in general.

13.6 Connecting back to vector spaces, and what’s still missing

A vector space with a norm — a length function \(\|v\|\) measuring a single vector’s magnitude, which Inner Product Spaces derives from an inner product — automatically becomes a metric space via \(d(u,v) = \|u-v\|\); both the Euclidean and taxicab distances above arise exactly this way from two different norms on \(\mathbb{R}^n\). But the discrete metric does not come from any norm on a vector space: it isn’t compatible with scaling (\(d(2x, 2y)\) should shrink or grow predictably under a norm-induced metric; the discrete metric ignores scaling entirely). That’s the generalization thread from the map again — metric spaces gain enormous generality (no arithmetic required at all) and lose the ability to talk about “scaling a distance,” which a vector space with a norm has for free.

13.7 What a metric space is NOT, and a callback

Not necessarily numeric or geometric-looking — the discrete metric is proof of that. And not automatically complete: a metric space is complete if every sequence that “ought to” converge (a Cauchy sequence — one whose terms get arbitrarily close to each other) actually does converge to a point in the space. This is exactly the gap Number Systems described between \(\mathbb{Q}\) and \(\mathbb{R}\): viewed as metric spaces under \(d(x,y)=|x-y|\), \(\mathbb{R}\) is complete and \(\mathbb{Q}\) is not — the sequence of rational truncations of \(\sqrt{2}\) is Cauchy in \(\mathbb{Q}\) but has no rational limit. Completeness, informally described back in chapter 6 as “no gaps,” now has a precise, general definition that applies to any metric space, not just \(\mathbb{R}\).

13.8 Where this leads

Topological Spaces takes the “open set” idea this chapter derived from distance and makes it primitive — dropping the requirement that closeness be measured by a number at all, which turns out to be exactly enough structure to keep talking about continuity and convergence in settings with no metric in sight.