Given a set and an element of , the singleton is that subset of whose only element is . Strictly speaking, we are considering here a singleton subset; we could also consider a singleton list or -tuple , but this is an equivalent concept.
Here, is classified by the characteristic map (where is the set of truth values) given by
As an injection to , is precisely the same map as itself is as a generalized element of ; the same goes for the -tuple as a map from . One can take this to justify the common abuse of notation (as it would normally be considered) in which or is written as when no confusion can result.
Note that the set of all singletons of elements of is isomorphic to itself. In this way, the entire concept can be seen as a triviality.
A subset of a singleton is called a subsingleton. In classical mathematics (using the principle of excluded middle), the only subsingletons are the singletons and the empty subset, but in constructive mathematics, this is an important concept.
Everything above can be generalised from the category of sets to any topos.
Last revised on April 6, 2017 at 00:15:04. See the history of this page for a list of all contributions to it.