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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:
In general, an n-dimensional CW complex is constructed by taking the disjoint union of a k-dimensional CW complex (for some <) with one or more copies of the n-dimensional ball. For each copy, there is a map that "glues" its boundary (the ( n − 1 ) {\displaystyle (n-1)} -dimensional sphere ) to elements of the k {\displaystyle k} -dimensional ...
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.
Let be the least uncountable ordinal.In an analog of Baire space derived from the -fold cartesian product of with itself, any closed set is the disjoint union of an -perfect set and a set of cardinality, where -closedness of a set is defined via a topological game in which members of are played.