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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 graph theory, a cograph, or complement-reducible graph, or P 4-free graph, is a graph that can be generated from the single-vertex graph K 1 by complementation and disjoint union. That is, the family of cographs is the smallest class of graphs that includes K 1 and is closed under complementation and disjoint union.
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.
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 : = /,
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.
The concept of disjoint union secretly underlies the above examples: the direct sum of abelian groups is the group generated by the "almost" disjoint union (disjoint union of all nonzero elements, together with a common zero), similarly for vector spaces: the space spanned by the "almost" disjoint union; the free product for groups is generated ...
Geometric join of two line segments.The original spaces are shown in green and blue. The join is a three-dimensional solid, a disphenoid, in gray.. In topology, a field of mathematics, the join of two topological spaces and , often denoted by or , is a topological space formed by taking the disjoint union of the two spaces, and attaching line segments joining every point in to every point in .