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.