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.

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.