Singleton (mathematics)

Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. This implies that a singleton is necessarily distinct from the element it contains, The ultrafilter lemma implies that non-principal ultrafilters exist on every infinite set (these are called ). Every net valued in a singleton subset X of is an ultranet in X. The Bell number integer sequence counts the number of partitions of a set (), if singletons are excluded then the numbers are smaller (). ==In category theory==

Structures built on singletons often serve as terminal objects or zero objects of various categories: • The statement above shows that the singleton sets are precisely the terminal objects in the category Set of sets. No other sets are terminal. • Any singleton admits a unique topological space structure (both subsets are open). These singleton topological spaces are terminal objects in the category of topological spaces and continuous functions. No other spaces are terminal in that category. • Any singleton admits a unique group structure (the unique element serving as identity element). These singleton groups are zero objects in the category of groups and group homomorphisms. No other groups are terminal in that category. ==Definition by indicator functions==

Let be a class defined by an indicator function b : X \to \{0, 1\}. Then is called a singleton if and only if there is some y \in X such that for all x \in X, b(x) = (x = y). ==Definition in Principia Mathematica==

The following definition was introduced in Principia Mathematica by Whitehead and Russell :\iota‘x = \hat{y}(y = x) Df. The symbol \iota‘x denotes the singleton \{x\} and \hat{y}(y = x) denotes the class of objects identical with x aka \{y : y=x\}. This occurs as a definition in the introduction, which, in places, simplifies the argument in the main text, where it occurs as proposition 51.01 (p. 357 ibid.). The proposition is subsequently used to define the cardinal number 1 as :1=\hat{\alpha}((\exists x)\alpha=\iota‘x) Df. That is, 1 is the class of singletons. This is definition 52.01 (p. 363 ibid.) ==See also==