A metric is supposed to measure closeness, but there is a strange way for closeness to disappear. Imagine a space in which every point can be recognized perfectly: two distinct points are always exactly one unit apart, no matter how similar they look or how they were named. In such a space, a sequence cannot creep toward a new point by making smaller and smaller moves. It must eventually land on the point and stay there. This is the world of the discrete metric.

The discrete metric is often the first example showing that the axioms of a [metric space](/page/Metric%20Space) allow more than geometric distance in Euclidean space. It converts any set into a metric space, but the resulting geometry is rigid: balls of radius less than $1$ isolate single points, every subset is open, compactness becomes finiteness in many cases, and continuous maps out of the space lose all local restrictions.

[example: A Sequence That Cannot Approach Without Arriving] Let $X$ be a set with at least two points, fix $x_0\in X$, and equip $X$ with the discrete metric: $d(x,x)=0$ and $d(x,y)=1$ when $x\ne y$. We show that a sequence $(x_n)_{n=1}^{\infty}$ cannot converge to $x_0$ unless it is eventually equal to $x_0$. Assume $x_n\to x_0$. By the definition of metric convergence with $\varepsilon=1/2$, there exists $N\in\mathbb{N}$ such that for every $n\ge N$, \begin{align*} d(x_n,x_0)<\frac{1}{2}. \end{align*} Fix such an $n\ge N$. In the discrete metric, exactly one of the following two cases holds. If $x_n=x_0$, then $d(x_n,x_0)=0$. If $x_n\ne x_0$, then $d(x_n,x_0)=1$. The second case is impossible, because \begin{align*} 1<\frac{1}{2} \end{align*} is false. Therefore the only possible case is $d(x_n,x_0)=0$, and by the defining property of the discrete metric this means $x_n=x_0$. Since this holds for every $n\ge N$, the sequence is eventually constant with value $x_0$. Thus, in the discrete metric, convergence to a point means that the sequence eventually arrives at that point and then stays there. [/example]

This example is the guiding phenomenon for the chapter. The discrete metric is not hard because its formula is complicated; it is important because it shows how much topology can be forced by a metric with only two distance values. It is the metric counterpart of the discrete topology, and it acts as a testing ground for definitions involving continuity, compactness, connectedness, and completeness.

## Definition

The guiding question is whether every set can be made into a metric space without imposing coordinates, order, algebra, or an ambient Euclidean space. The construction we need should distinguish equality from non-equality and refuse to measure any finer relationship between distinct points.

[definition: Discrete Metric] Let $X$ be a set. The discrete metric on $X$ is the function $d:X\times X\to\mathbb{R}$ defined by $d(x,x)=0$ for every $x\in X$ and $d(x,y)=1$ whenever $x,y\in X$ satisfy $x\ne y$. [/definition]

The formula is only useful if it really behaves like a distance. Positivity and symmetry follow directly from the two cases in the definition, but the triangle inequality requires checking that a distance equal to $1$ cannot be forced to be less than the sum of two zero distances. That is the only possible obstruction to turning this equality test into a metric.

[quotetheorem:8358]

## Scaling and Balls

This theorem gives a metric space structure on every set. The construction has a visible scale equal to $1$, so the next definition records the same idea with an arbitrary positive separation size. This is useful when numerical diameters or Lipschitz constants are being compared.

[definition: Scaled Discrete Metric] Let $X$ be a set and let $a>0$. The scaled discrete metric with scale $a$ is the function $d_a:X\times X\to\mathbb{R}$ defined by $d_a(x,x)=0$ for every $x\in X$ and $d_a(x,y)=a$ whenever $x,y\in X$ satisfy $x\ne y$. [/definition]

All scaled discrete metrics determine the same open sets, but they do not have the same numerical diameter when $X$ has at least two points. The scale records a chosen unit of separation, while the topology records only that distinct points are separated.

The metric formula is so short that its topology can be read directly from the balls. Before discussing open sets, we need the concrete ball computation because it is the mechanism behind almost every later theorem on convergence, compactness, and continuity.

[example: Balls in a Discrete Metric Space] Let $(X,d)$ be a set with the discrete metric, fix $x\in X$, and let $r>0$. We compute the open ball \begin{align*} B(x,r)=\{y\in X:d(x,y)<r\}. \end{align*} First suppose $0<r\le 1$. Since $d(x,x)=0$ and $0<r$, we have \begin{align*} d(x,x)=0<r, \end{align*} so $x\in B(x,r)$. If $y\ne x$, then the definition of the discrete metric gives $d(x,y)=1$. Because $r\le 1$, the inequality $1<r$ is false, so $d(x,y)<r$ is false. Hence no point $y\ne x$ belongs to $B(x,r)$, and therefore \begin{align*} B(x,r)=\{x\}. \end{align*} Now suppose $r>1$. For $y\in X$, either $y=x$ or $y\ne x$. If $y=x$, then \begin{align*} d(x,y)=d(x,x)=0<r. \end{align*} If $y\ne x$, then \begin{align*} d(x,y)=1<r. \end{align*} In both cases $y\in B(x,r)$, so every point of $X$ lies in the ball. Thus \begin{align*} B(x,r)=X. \end{align*} The radius $1$ is the threshold: balls of radius at most $1$ isolate the centre, while balls of radius greater than $1$ contain the whole space. [/example]

## The Discrete Topology Generated by the Metric

A metric space generates a topology by declaring unions of open balls to be open. For the Euclidean metric, this produces a topology with local shape and boundary behavior. For the discrete metric, the balls of radius $1/2$ already isolate individual points, so the resulting topology is as fine as possible.

To make that comparison precise, we first need the general definition of the topology associated with a metric. This definition is the bridge from numerical distances to topological language such as [open set](/page/Open%20Set), closure, compactness, and connectedness.

[definition: Metric Topology] Let $(X,d)$ be a metric space. The metric topology induced by $d$ is the collection $\tau_d$ of subsets $U\subset X$ such that for every $x\in U$ there exists $r>0$ with $B(x,r) \subset U$. [/definition]

The metric topology records which subsets are visible through open balls. In the discrete metric, every singleton is visible at scale $1/2$. We therefore need a name for the topology in which every possible subset is already open.