Disjoint union (topology)

Let {X_i : i ∈ I} be a family of topological spaces indexed by I. Let

${\displaystyle X=\coprod _{i}X_{i}}$

be the disjoint union of the underlying sets. For each i in I, let

${\displaystyle \varphi _{i}:X_{i}\to X\,}$

be the canonical injection (defined by ${\displaystyle \varphi _{i}(x)=(x,i)}$). The disjoint union topology on X is defined as the finest topology on X for which all the canonical injections ${\displaystyle \varphi _{i}}$ 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 preimage ${\displaystyle \varphi _{i}^{-1}(U)}$ 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.

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:

Characteristic property of disjoint unions

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.

If each X_i 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.

• Every disjoint union of discrete spaces is discrete
• Separation
  • Every disjoint union of T₀ spaces is T₀
  • Every disjoint union of T₁ spaces is T₁
  • Every disjoint union of Hausdorff spaces is Hausdorff
• Connectedness
  • The disjoint union of two or more nonempty topological spaces is disconnected

• 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