nLab singleton

Idea

Given a set $X$ and an element $a$ of $X$, the singleton $\{a\}$ is that subset of $X$ whose only element is $a$. Strictly speaking, we are considering here a singleton subset; we could also consider a singleton list or $1$-tuple $(a)$, but this is an equivalent concept.

Here, $\{a\}$ is classified by the characteristic map $c: X \to \Omega$ (where $\Omega$ is the set of truth values) given by

$$c(b) = (a = b) .$$

As an injection to $X$, $\{a\}$ is precisely the same map $1 \to X$ as $a$ itself is as a generalized element of $X$; the same goes for the $1$-tuple $(a)$ as a map from $[1]$. One can take this to justify the common abuse of notation (as it would normally be considered) in which $\{a\}$ or $(a)$ is written as $a$ when no confusion can result.

Note that the set of all singletons of elements of $X$ is isomorphic to $X$ itself. In this way, the entire concept can be seen as a triviality.

A subset of a singleton is called a subsingleton. In classical mathematics (using the principle of excluded middle), the only subsingletons are the singletons and the empty subset, but in constructive mathematics, this is an important concept.

Everything above can be generalised from the category of sets to any topos.

Singleton subsets are important in distinguishing between two kinds of categorical set theories; there are the categorical set theories like ETCS where elements are singleton subsets; and then there are the categorical set theories like structural ZFC where elements are different from singleton subsets but have a reflection into singleton subsets.

Related concepts

• point, point space

• empty set