singleton

Given a set $X$ and an element $a$ of $X$, the **singleton** $\{a\}$ is that subset of $X$ whose only element is $a$.

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 $X$. One can take this to justify the common abuse of notation (as it would normally be considered) in which $\{a\}$ 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.

A subset of a singleton is called a *subsingleton*.

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

Revised on September 4, 2010 18:55:56
by Toby Bartels
(173.190.156.19)