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In mathematics, the disjoint union (or discriminated union) of the sets A and B is the set formed from the elements of A and B labelled (indexed) with the name of the set from which they come. So, an element belonging to both A and B appears twice in the disjoint union, with two different labels.
The disjoint union space X, together with the canonical injections, can be characterized by the following universal property: If Y is a topological space, and f i : X i → Y is a continuous map for each i ∈ I, then there exists precisely one continuous map f : X → Y such that the following set of diagrams commute:
A wedge sum of two circles. In topology, the wedge sum is a "one-point union" of a family of topological spaces.Specifically, if X and Y are pointed spaces (i.e. topological spaces with distinguished basepoints and ) the wedge sum of X and Y is the quotient space of the disjoint union of X and Y by the identification : = /,
λ-system (Dynkin system) – Family closed under complements and countable disjoint unions; π-system – Family of sets closed under intersection; Ring of sets – Family closed under unions and relative complements; Russell's paradox – Paradox in set theory (or Set of sets that do not contain themselves)
The best mathematical definition of disjoint union is to be a coproduct in the category of sets. As such, the discrete union is defined up to an isomorphism, and the definition with "index space" given in the article is just one realization among others. When the sets are pairwise disjoint, the usual union is another realization.
The pushout of f and g is the disjoint union of X and Y, where elements sharing a common preimage (in Z) are identified, together with the morphisms i 1, i 2 from X and Y, i.e. = / where ~ is the finest equivalence relation (cf. also this) such that f(z) ~ g(z) for all z in Z.
The disjoint union topology is the final topology with respect to the inclusion maps. The final topology is also the topology that every direct limit in the category of topological spaces is endowed with, and it is in the context of direct limits that the final topology often appears.
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French bord, giving cobordism) of a manifold. Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher.