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In topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. The name coproduct originates from the fact that the disjoint union is the categorical dual of the product space construction.
Definition Let be a family of topological spaces indexed by I. Let be the disjoint union of the underlying sets. For each i in I, let be the canonical injection. The disjoint union topology on X is defined as the finest topology on X for which the canonical injections are continuous (i.e. the final topology for the family of functions ). Explicitly, the disjoint union topology can be described as follows. A subset U of X is open in X if and only if its preimage is open in Xi for each i ∈ I. Properties 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 fi Xi → Y is a continuous map for each i ∈ I, then there exists precisely one continuous map fX → Y such that the following set of diagrams commute: This shows that the disjoint union is the coproduct in the category of topological spaces. It follows from the above universal property that a map f X → Y is continuous iff fi = f o φi is continuous for all i in I. In addition to being continuous, the canonical injections φi Xi → X are open and closed maps. It follows that the injections are topological embeddings so that each Xi may be canonically thought of as a subspace of X. Examples If each Xi is homeomorphic to a fixed space A, then the disjoint union X will be homeomorphic to A × I where I is given the discrete topology. Preservation of topological properties See also | ||||||||
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