# Definition:Singleton

## Definition

A singleton is a set that contains exactly one element.

The singleton containing only the element $a$ can be written $\left\{{a}\right\}$.

When this happens, we must be careful to distinguish between the element itself, that is, $a$, and the set containing it, that is, $\left\{{a}\right\}$.

The set $\left\{{a}\right\}$ is known as the singleton of $a$.

## Formal Definition

The concept of the singleton set can be formalized rigorously as:

$\left\{{A}\right\} := \left\{{x : x = A}\right\}$

With this definition, the singleton of proper classes is equal to the empty set.

## Also known as

Some authors use the term unit set.

## Also see

• Results about singletons can be found here.