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In [[General topology|general topology]] and related areas of [[Mathematics|mathematics]], the '''disjoint union''' (also called the  '''direct sum''', '''free union''', '''free sum''', '''topological sum''', or '''coproduct''') of a [[Indexed family|family]] of [[Topological space|topological space]]s is a space formed by equipping the [[Disjoint union|disjoint union]] of the underlying sets with a [[Natural topology|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''<sub>''i''</sub> : ''i'' ∈ ''I''} be a family of topological spaces indexed by ''I''. Let
:<math>X = \coprod_i X_i</math>
be the [[Disjoint union|disjoint union]] of the underlying sets. For each ''i'' in ''I'', let
:<math>\varphi_i : X_i \to X\,</math>
be the '''canonical injection''' (defined by <math>\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>\varphi_i</math> are continuous (i.e.: it is the [[Final topology|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 set|open]] in ''X'' [[If and only if|if and only if]] its preimage <math>\varphi_i^{-1}(U)</math> is open in ''X''<sub>''i''</sub> for each ''i'' ∈ ''I''. Yet another formulation is that a subset ''V'' of ''X'' is open relative to ''X'' iff its intersection with ''X<sub>i</sub>'' is open relative to ''X<sub>i</sub>'' for each ''i''.

==Properties==
The disjoint union space ''X'', together with the canonical injections, can be characterized by the following [[Universal property|universal property]]: If ''Y'' is a topological space, and ''f<sub>i</sub>'' : ''X<sub>i</sub>'' → ''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 [[Commutative diagram|commute]]:
[[Image:Coproduct-02.svg|center|Characteristic property of disjoint unions]]
This shows that the disjoint union is the [[Coproduct|coproduct]] in the [[Category of topological spaces|category of topological spaces]]. It follows from the above universal property that a map ''f'' : ''X'' → ''Y'' is continuous iff ''f<sub>i</sub>'' = ''f'' o φ<sub>''i''</sub> is continuous for all ''i'' in ''I''.

In addition to being continuous, the canonical injections φ<sub>''i''</sub> : ''X''<sub>''i''</sub> → ''X'' are [[Open and closed maps|open and closed maps]]. It follows that the injections are topological embeddings so that each ''X''<sub>''i''</sub> may be canonically thought of as a subspace of ''X''.

==Examples==
If each ''X''<sub>''i''</sub> is homeomorphic to a fixed space ''A'', then the disjoint union ''X'' is homeomorphic to the product space ''A'' &times; ''I'' where ''I'' has the [[Discrete space|discrete topology]].

==Preservation of topological properties==
* Every disjoint union of [[Discrete space|discrete space]]s is discrete
*''Separation''
** Every disjoint union of T<sub>0</sub> spaces is T<sub>0</sub>
** Every disjoint union of [[T1 space|T<sub>1</sub> space]]s is T<sub>1</sub>
** Every disjoint union of [[Hausdorff space]]s is Hausdorff
*''Connectedness''
** The disjoint union of two or more nonempty topological spaces is disconnected

==See also==
* [[Product topology|product topology]], the dual construction
* [[Subspace topology|subspace topology]] and its dual quotient topology
* topological union, a generalization to the case where the pieces are not disjoint

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[[Category:General topology]]

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