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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 space]]s 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.
 
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 (set theory)|family]] of topological spaces indexed by ''I''. Let
:<math>X = \coprod_i X_i</math>
be the [[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 [[largest topology]] on ''X'' for which the canonical injections are [[continuous function (topology)|continuous]] (i.e. the [[final topology]] for the family of functions {φ<sub>''i''</sub>}).
 
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]] 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]]: 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.png|center|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<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]]. It follows that the injections are [[topological embedding]]s so that each ''X''<sub>''i''</sub> may be canonically thought of as a [[subspace (topology)|subspace]] of ''X''.
 
==Examples==
If each ''X''<sub>''i''</sub> is [[homeomorphic]] to a fixed space ''A'', then the disjoint union ''X'' will be homeomorphic to ''A'' &times; ''I'' where ''I'' is given the [[discrete topology]].
 
==Preservation of topological properties==
* every disjoint union of [[discrete space]]s is discrete
*''Separation''
** every disjoint union of [[T0 space|T<sub>0</sub> space]]s 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 (topology)|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
 
{{DEFAULTSORT:Disjoint Union (Topology)}}
[[Category:General topology]]

Latest revision as of 00:08, 14 February 2014

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