Editing Singleton (mathematics) (section)

==In category theory==

Structures built on singletons often serve as [[terminal object]]s or [[zero object]]s of various [[Category (category theory)|categories]]:
* The statement above shows that the singleton sets are precisely the terminal objects in the category '''[[Category of sets|Set]]''' of [[Set (mathematics)|set]]s. 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 function]]s. No other spaces are terminal in that category.
* Any singleton admits a unique [[Group (mathematics)|group]] structure (the unique element serving as [[identity element]]). These singleton groups are [[Initial object|zero object]]s in the category of groups and [[group homomorphism]]s. No other groups are terminal in that category.