Disjoint union (topology)


In general topology and related areas of mathematics, the disjoint union 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 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 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 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.

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 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 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 is homeomorphic to the product space A × I where I has the discrete topology.

Preservation of topological properties