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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 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.
In graph theory, a branch of mathematics, the disjoint union of graphs is an operation that combines two or more graphs to form a larger graph. It is analogous to the disjoint union of sets , and is constructed by making the vertex set of the result be the disjoint union of the vertex sets of the given graphs, and by making the edge set of the ...
Often replaced by a horizontal bar. For example, 3 / 2 or . 2. Denotes a quotient ... Used for the disjoint union of a family of sets, ...
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.
In this way, the disjoint union construction provides a way of viewing any family of sets indexed by as a set "fibered" over , and conversely, for any set : fibered over , we can view it as the disjoint union of the fibers of . Jacobs has referred to these two perspectives as "display indexing" and "pointwise indexing".
The continuous maps h : X ∪ f Y → Z are in 1-1 correspondence with the pairs of continuous maps h X : X → Z and h Y : Y → Z that satisfy h X (f(a))=h Y (a) for all a in A.. In the case where A is a closed subspace of Y one can show that the map X → X ∪ f Y is a closed embedding and (Y − A) → X ∪ f Y is an open embedding.
If x and y are topologically distinguishable, then the singleton sets {x} and {y} must be disjoint. On the other hand, if the singletons {x} and {y} are separated, then the points x and y must be topologically distinguishable. Thus for singletons, topological distinguishability is a condition in between disjointness and separatedness.