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In category theory the disjoint union is defined as a coproduct in the category of sets. As such, the disjoint union is defined up to an isomorphism, and the above definition is just one realization of the coproduct, among others. When the sets are pairwise disjoint, the usual union is another realization of the coproduct.
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:
Two disjoint sets. In set theory in mathematics and formal logic, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. [1] For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of two ...
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 ...
The disjoint union of the above two examples is locally compact in sense (1) but not in senses (2), (3) or (4). The right order topology on the real line is locally compact in senses (1) and (3) but not in senses (2) or (4), because the closure of any neighborhood is the entire non-compact space.
A 1-dimensional CW complex is constructed by taking the disjoint union of a 0-dimensional CW complex with one or more copies of the unit interval. For each copy, there is a map that "glues" its boundary (its two endpoints) to elements of the 0-dimensional complex (the points).
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 .
The 2-regular graphs are the disjoint unions of cycle graphs. [4] More generally, every graph is the disjoint union of connected graphs, its connected components. The cographs are the graphs that can be constructed from single-vertex graphs by a combination of disjoint union and complement operations. [5]