In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, 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. Roughly speaking, in the disjoint union the given spaces are considered as part of a single new space where each looks as it would alone and they are isolated from each other.
The name coproduct originates from the fact that the disjoint union is the categorical dual of the product space construction.
Let {Xi : i ∈ I} 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 (defined by ). The disjoint union topology on X is defined as the finest topology on X for which all the canonical injections are continuous (i.e.: it is the final topology on X induced by the canonical injections).
Explicitly, the disjoint union topology can be described as follows. A subset U of X is open in X if and only if its is open in Xi for each i ∈ I. Yet another formulation is that a subset V of X is open relative to X iff its intersection with Xi is open relative to Xi for each i.
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 f : X → Y such that the following set of diagrams commute:
This shows that the disjoint union is the coproduct in the . 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.
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In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, 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. Roughly speaking, in the disjoint union the given spaces are considered as part of a single new space where each looks as it would alone and they are isolated from each other.
In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set with respect to a family of functions from topological spaces into is the finest topology on that makes all those functions continuous. The quotient topology on a quotient space is a final topology, with respect to a single surjective function, namely the quotient map. The disjoint union topology is the final topology with respect to the inclusion maps.
In mathematics, the category of topological spaces, often denoted Top, is the whose s are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of topological spaces using the techniques of is known as categorical topology. N.B. Some authors use the name Top for the categories with topological manifolds, with compactly generated spaces as objects and continuous maps as morphisms or with the .
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