Disjoint union (topology)

From Wikipedia, the free encyclopedia

In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, 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.

1 Definition
2 Properties
3 Examples
4 Preservation of topological properties
5 See also

[edit] Definition

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

$X = \coprod_i X_i$

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

$\varphi_i : X_i \to X\,$

be the canonical injection. The disjoint union topology on X is defined as the finest topology on X for which the canonical injections are continuous (i.e. the final topology for the family of functions {φ_i}).

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 $\varphi_i^{-1}(U)$ is open in X_i for each i ∈ I.

[edit] 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:

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.

[edit] 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.

[edit] Preservation of topological properties

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 topological spaces is disconnected