singleton in nLab

Contents

Contents

Idea

Given a set XX and an element aa of XX, the singleton {a}\{a\} is that subset of XX whose only element is aa. Strictly speaking, we are considering here a singleton subset; we could also consider a singleton list or 11-tuple (a)(a), but this is an equivalent concept.

Here, {a}\{a\} is classified by the characteristic map c:X→Ωc: 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 1→X1 \to X as aa itself is as a generalized element of XX; the same goes for the 11-tuple (a)(a) as a map from [1][1]. One can take this to justify the common abuse of notation (as it would normally be considered) in which {a}\{a\} or (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. 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.

Singleton subsets are important in distinguishing between two kinds of categorical set theories; there are the categorical set theories like ETCS where elements are singleton subsets; and then there are the categorical set theories like structural ZFC where elements are different from singleton subsets but have a reflection into singleton subsets.

• point, point space

• empty set