Singleton (mathematics)

${\displaystyle X}$

${\displaystyle X.}$

The

• 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
  zero objects in the category of groups and group homomorphisms
  . No other groups are terminal in that category.

Let S be a class defined by an indicator function

$${\displaystyle b:X\to \{0,1\}.}$$

${\displaystyle y\in X}$

${\displaystyle x\in X,}$

$${\displaystyle b(x)=(x=y).}$$

Definition in Principia Mathematica

The following definition was introduced by Whitehead and Russell^[3]

${\displaystyle \iota }$‘${\displaystyle x={\hat {y}}(y=x)}$ Df.

The symbol ${\displaystyle \iota }$‘${\displaystyle x}$ denotes the singleton ${\displaystyle \{x\}}$ and ${\displaystyle {\hat {y}}(y=x)}$ denotes the class of objects identical with ${\displaystyle x}$ aka ${\displaystyle \{y:y=x\}}$. 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

${\displaystyle 1={\hat {\alpha }}((\exists x)\alpha =\iota }$‘${\displaystyle x)}$ Df.

That is, 1 is the class of singletons. This is definition 52.01 (p.363 ibid.)

See also

• Class (set theory) – Collection of sets in mathematics that can be defined based on a property of its members
• Isolated point – Point of a subset S around which there are no other points of S
• Uniqueness quantification – Logical property of being the one and only object satisfying a condition
• Urelement – Concept in set theory

References

• OCLC 945169917
  .