Disjoint union (topology)
This article does not cite any sources. (October 2009) (Learn how and when to remove this template message)

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, two or more spaces may be considered together, each looking as it would alone.
The name coproduct originates from the fact that the disjoint union is the categorical dual of the product space construction.
Definition
Let {X_{i} : 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.
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 X_{i} for each i ∈ I.
Yet another formulation is that a subset V of X is open relative to X iff its intersection with X_{i} is open relative to X_{i} for each 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 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:
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 f_{i} = f o φ_{i} is continuous for all i in I.
In addition to being continuous, the canonical injections φ_{i} : X_{i} → X are open and closed maps. It follows that the injections are topological embeddings so that each X_{i} may be canonically thought of as a subspace of X.
Examples
If each X_{i} 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
 every disjoint union of discrete spaces is discrete

Separation
 every disjoint union of T_{0} spaces is T_{0}
 every disjoint union of T_{1} spaces is T_{1}
 every disjoint union of Hausdorff spaces is Hausdorff

Connectedness
 the disjoint union of two or more nonempty topological spaces is disconnected
See also
 product topology, the dual construction
 subspace topology and its dual quotient topology
 topological union, a generalization to the case where the pieces are not disjoint