Given a set XX and an element aa of XX, the singleton {a}\{a\} is that subset of XX whose only element is aa.

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

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

As an injection to XX, {a}\{a\} is precisely the same map 1X1 \to X as aa itself is as a generalized element of XX. One can take this to justify the common abuse of notation (as it would normally be considered) in which {a}\{a\} is written as aa when no confusion can result.

Note that the set of all singletons of elements of XX is isomorphic to XX 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 (