nLab
discrete space

Idea

A discrete space is, in general, an object of a concrete category of spaces that is free on its underlying set. Every topological concrete category has discrete spaces; so do many other categories, such as Diff and Haus Top.

Definition

For this to make sense, we want the functor

$$\mathrm{Set}\stackrel{F}{\to }\mathrm{Sp}\stackrel{U}{\to }\mathrm{Set}$$Set \stackrel{F}\to Sp \stackrel{U}\to Set

to be (naturally isomorphic to) the identity functor on Set, where $F:\mathrm{Set}\to \mathrm{Sp}$ is left adjoint to the forgetful functor $U:\mathrm{Sp}\to \mathrm{Set}$. (This is true, for example, if $\mathrm{Sp}$ is Top, Diff, Loc, etc, although $\mathrm{Loc}$ is not quite a concrete category since the forgetful functor from locales is not faithful.)

Assuming that $U$ is faithful (as it should be when $\mathrm{Sp}$ is a concrete category), we can characterise a discrete space $X$ as one such that every function from $X$ to $Y$ (for $Y$ any space) is a morphism of spaces. (More precisely, this means that every function from $U(X)$ to $U(Y)$ is the image under $U$ of a morphism from $X$ to $Y$.)

A topological space is discrete in the sense above only if the diagonal map $X\times X\to X$ is open; the converse holds if $X$ satisfies the $T_0$ separation axiom. In locale theory, the condition that a locale is discrete may be split into two parts: that $X\times X\to X$ is open and that $X\to 1$ is open. A locale that satisfies the latter condition is called overt; note that every locale is $T_0$ while every topological space is overt. In Abstract Stone Duality, a space is called discrete if $X\times X\to X$ is open, which corresponds to the existence of an equality relation on $X$; discrete spaces as described above correspond to discrete overt spaces in ASD.

Examples

The best known example is a discrete topological space, that is one, $X$, in which all subsets of $X$ are open in the topology. This same space serves as a discrete object in many subcategories and supercategories of $\mathrm{Top}$, from convergence spaces (where the only proper filter that converges to a point is the free ultrafilter at that point) to (say) paracompact Hausdorff spaces (because a discrete topological space has those properties). It is also sobre and thus serves as a discrete locale, whose corresponding frame is the power set of $X$; see CABA.

A discrete uniform space $X$ has all reflexive relation as entourages, or equivalently all covers as uniform covers. It is the only uniformity (on a given set) whose underlying topology is discrete.

Strictly speaking, there is no discrete metric space on any set with more than one element, because the forgetful functor has no left adjoint. However, there is a discrete extended metric space, given by $d(x,y)=\mathrm{\infty }$ whenever $x\ne y$. More usually, the term ‘discrete metric’ is used when $d(x,y)=1$ for $x\ne y$, which is discrete in the category of metric spaces of diameter at most $1$. (Comparing the adjoint functor theorem, the problem with $\mathrm{Met}$ is that it generally lacks infinitary products; in contrast, $\mathrm{Ext}\mathrm{Met}$ and ${\mathrm{Met}}_1$ are complete.)