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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. Roughly speaking, two or more spaces may be considered together, each looking as it would alone.## Definition

_{i}}).## Properties

## Examples

## Preservation of topological properties

## See also

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

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

- $X\; =\; coprod\_i\; X\_i$

- $varphi\_i\; :\; X\_i\; to\; X,$

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.

Yet another formulation is that a subset V of X is open relative to X iff if 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:

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 will be homeomorphic to A × I where I is given the discrete topology.

- 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
- Connectedness
- the disjoint union of two or more 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

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Last updated on Saturday August 23, 2008 at 03:29:36 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday August 23, 2008 at 03:29:36 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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