Disjoint union (topology)

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.

The name coproduct originates from the fact that the disjoint union is the categorical dual of the product space construction.

are continuous (i.e.: it is the final topology on X induced by the canonical injections).

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.

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.