Metric Spaces — Mathematics Knowledge System

Overview

A metric space is a set X equipped with a distance function d: X × X → ℝ that assigns a nonnegative real number to every pair of points, measuring how 'far apart' they are. This distance function, called a metric, must satisfy three natural axioms: identical points have distance zero (and only identical points do), distance is symmetric, and distances satisfy a triangle inequality. Metric spaces give the most familiar route into topology, because every metric induces a topology — a collection of open sets built from open balls — recovering the usual ε-neighborhood notion of closeness. Many different metrics can induce the very same topology, which is exactly the sense in which topology is a strictly more general framework than metric geometry.

Intuition

Imagine measuring distances on a map with a tape measure: between any two cities you get a nonnegative number, the number is the same whether you measure from city A to city B or back, and a direct route is never longer than a detour through some third city — that detour rule is the triangle inequality. A metric is just this idea made abstract: any rule d(x,y) satisfying those three sensible properties turns any set into a space where 'distance' makes sense, whether the points are cities, functions, or strings of text. Once you can measure distance, you get open disks (balls) around each point for free, and declaring a set 'open' whenever every one of its points has a wiggle-room ball entirely inside it recovers the whole topological structure without extra work.

Formal Definition

Definition

A metric on a set X is a function d: X × X → ℝ satisfying, for all x, y, z ∈ X:

d(x,y) \geq 0, \text{ and } d(x,y) = 0 \iff x = y

Distance is nonnegative and only zero between a point and itself

Identity of indiscernibles

d(x,y) = d(y,x)

The distance from x to y equals the distance from y to x

Symmetry

d(x,z) \leq d(x,y) + d(y,z)

Going directly from x to z is never longer than detouring through y

Triangle inequality

B(x,\varepsilon) = \{ y \in X : d(x,y) < \varepsilon \}

The set of points within distance ε of x; open balls are the building blocks of the metric topology

Open ball

\tau_d = \{ U \subseteq X : \forall x \in U,\ \exists \varepsilon>0 \text{ with } B(x,\varepsilon) \subseteq U \}

The topology generated by d: a set is open iff every point has an open ball around it fully contained in the set

Induced topology

Notation

Notation	Meaning
$(X,d)$	A metric space: a set X together with a metric d
$B(x,\varepsilon)$	The open ball of radius ε centered at x
$\tau_d$	The topology induced on X by the metric d

Properties

Euclidean metric

d(x,y) = \sqrt{\sum_{i=1}^n (x_i-y_i)^2}

Example: The ordinary straight-line distance in ℝⁿ; induces the standard topology on ℝⁿ.

Taxicab (Manhattan) metric

d(x,y) = \sum_{i=1}^n |x_i - y_i|

Example: Distance measured along grid-aligned streets in ℝⁿ; a different formula from the Euclidean metric but induces the same topology on ℝⁿ.

Discrete metric

d(x,y) = 0 \text{ if } x=y, \text{ else } 1

Example: Every subset is open (each singleton is a ball of radius <1), so the induced topology is the discrete topology on X.

Metrizability

\text{A topological space } (X,\tau) \text{ is metrizable if there exists a metric } d \text{ with } \tau_d = \tau.

Condition: Not every topological space comes from a metric — e.g. non-Hausdorff spaces like the indiscrete topology on a set with ≥2 points are never metrizable.

Worked Examples

Compute the direct distance from x to z.
$d(1,8) = |1-8| = 7$
Compute the detoured sum through y=5.
$d(1,5) + d(5,8) = 4 + 3 = 7$
Compare: 7 ≤ 7 holds (with equality, since 5 lies between 1 and 8).
$d(1,8) \leq d(1,5)+d(5,8)$

Answer: The triangle inequality holds, with equality since y=5 lies on the segment between x=1 and z=8.

Practice Problems

Difficulty 4/10

Which property is NOT one of the metric axioms?

d(x,y) ≥ 0d(x,y) = d(y,x)d(x,z) ≤ d(x,y) + d(y,z)d(x,y) = d(x,z) for all z

Difficulty 5/10

In the discrete metric on a set X with |X| ≥ 2, what is the open ball B(x, 1)?

Difficulty 7/10

Prove that every open ball B(x,ε) in a metric space is an open set in the induced topology.

Quiz

Which of the following defines the topology induced by a metric d on X?

The collection of all subsets of XU is open iff every point of U has an open ball around it contained in UU is open iff U is a single open ballU is open iff d(x,y)=0 for all x,y in U

Two different metrics on the same set X can induce:

Always different topologiesThe same topologyNever a topology at allOnly the discrete topology

Summary

A metric space (X,d) is a set with a distance function satisfying nonnegativity/identity of indiscernibles, symmetry, and the triangle inequality.
Open balls B(x,ε) = {y : d(x,y)<ε} generate the induced topology τ_d: a set is open iff every point has a ball around it inside the set.
Different metrics (Euclidean, taxicab, discrete) can induce the same or different topologies — metrizability is a strictly stronger property than being merely a topological space.

References

WebsiteWikipedia — Metric space