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Disjoint union (topology)

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From Wikipedia, the free encyclopedia
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  1. Disjoint union (topology)

In general topology and related areas of mathematics, the disjoint union1 (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.

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  • Definition
  • Properties
  • Examples
  • Preservation of topological properties
  • See also
  • References
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Contents
Contents8 sections
  • (Top)
  • Definition
  • Properties
  • Examples
  • Preservation of topological properties
  • See also
  • References
  • Recent changes
  • Rabbit hole

Definition

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

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

be the canonical injection (defined by ). The disjoint union topology on X is defined as the finest topology on X for which all the canonical injections 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 is open in Xi for each i ∈ I. Yet another formulation is that a subset V of X is open relative to X iff its intersection with Xi is open relative to Xi for each 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 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:

Characteristic property of disjoint unions
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 fi = f o φi is continuous for all i in I.

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.

Examples

If each Xi 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 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

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

References

  1. 1"disjoint union topological space in nLab". ncatlab.org. Retrieved 5 May 2025.
Study aids for Disjoint union (topology)

Terminology

disjoint union
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 topol…
direct sum
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 topol…
free union
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 topol…
free sum
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 topol…
topological sum
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 topol…
coproduct
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 topol…
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 topol…
canonical injection
be the canonical injection (defined by ).

Terms the article defines in bold, quoted in context.

Categories:
  • General topology

458 words · 1 references · revision 1377316444 · CC BY-SA 4.0

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Key concepts

  • Disjoint union2
  • Product space2
  • General topology1
  • Indexed family1
  • Topological space1
  • Natural topology1
  • Categorical dual1
  • Iff2
  • Finest topology1
  • Continuous function (topology)1

The articles this one links to most, weighted towards the introduction.

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Created
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59 revisions in total · 41 contributors

  1. 28 Sept 2026, 21:43 UTCLemondogem§ Definition Avoid misuse of desclist formatting to indent math by using the `display=block` option of <math> instead+30diff
  2. 3 Jun 2025, 19:05 UTCGoldRomeanAdding local short description: "Mathematical term", overriding Wikidata description "space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology"+40diff
  3. 7 May 2025, 03:07 UTCBattyBot§ top Replaced {{unreferenced}} with {{more citations needed}} and other General fixes+5diff
  4. 5 May 2025, 14:01 UTCJoeNMLC§ top add cite+184diff
  5. 24 May 2024, 15:37 UTCBoleynStill needs refs+28diff
  6. 15 Jun 2023, 17:10 UTCHbghlyjpng to svg0diff
  7. 27 Apr 2021, 23:12 UTCAxelBoldt§ Definition link final topology+82diff
  8. 4 Mar 2021, 16:34 UTCAxelBoldtbetter "rough description"+79diff

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Created
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