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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.
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 .
If {X i} is a collection of spaces and X is the (set-theoretic) disjoint union of {X i}, then the coproduct topology (or disjoint union topology, topological sum of the X i) on X is the finest topology for which all the injection maps are continuous. Core-compact space Cosmic space A continuous image of some separable metric space. [3 ...
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 ...
One can form the topological union of an arbitrary family of topological spaces as above, but if the topologies do not agree on the intersections then the inclusions will not necessarily be embeddings. One can also describe the topological union by means of the disjoint union.
It follows that a locally connected space X is a topological disjoint union of its distinct connected components. Conversely, if for every open subset U of X, the connected components of U are open, then X admits a base of connected sets and is therefore locally connected. [23]
This realization of a combinatorial graph as a topological space is sometimes called a topological graph. 3-regular graphs can be considered as generic 1-dimensional CW complexes. Specifically, if X is a 1-dimensional CW complex, the attaching map for a 1-cell is a map from a two-point space to X , f : { 0 , 1 } → X {\displaystyle f:\{0,1 ...