Unitary set

In mathematics, a unitary group is a set with a single element. For example, the set {0} is a unitary set. Note that a set such as, for example, { {1; 1} } is also a unitary set: the only element is a set. A set is unitary if and only if its cardinality is one. In the construction -theortic-set of natural numbers, the number 1 is defined as the unitary set {0}. In the axiomatic theory of sets, the existence of unitary assemblies is a consequence of the axiom of the empty set and axiom of mating: the first gives void, and the last, applied to the pairing of { } and { }, produces the unitary set {{!!{displaystyle {{{{}}}}Yeah. A is a set and S is any unit set, there is exactly one function A a S, the constant function that sends each element of A element S. Structures built on unit assemblies often serve as terminal or end objects or zero objects of various categories:

• The previous statement shows that each unitary group S is a terminal object in Setthe category of sets and functions. There are no other terminal sets in that category.

• Any unit set can be presented as a topological space in a single form (all subsets are open, that is, only empty and unitary set: the same as empty, discreet and indiscreet space at the same time). These topological spaces on a unitary set are terminal objects in the Top category of the topological spaces and continuous functions. There are no other kind of terminal spaces in that category.

• Any unit set can be presented as a group in one form (the only element as an identity). these groups on a unitary set are the zero objects in the Grp category of groups and homomorphisms. There are no other zero objects in that category.