# 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**. Roughly speaking, in the disjoint union the given spaces are considered as part of a single new space where each looks as it would alone and they are isolated from each other.

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

- [math]\displaystyle{ X = \coprod_i X_i }[/math]

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

- [math]\displaystyle{ \varphi_i : X_i \to X\, }[/math]

be the **canonical injection** (defined by [math]\displaystyle{ \varphi_i(x)=(x,i) }[/math]). The **disjoint union topology** on *X* is defined as the finest topology on *X* for which all the canonical injections [math]\displaystyle{ \varphi_i }[/math] 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 [math]\displaystyle{ \varphi_i^{-1}(U) }[/math] 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*for each

_{i}*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* is homeomorphic to the product space *A* × *I* where *I* has 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

- Every disjoint union of T
*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

Original source: https://en.wikipedia.org/wiki/Disjoint union (topology).
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