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189:. 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.
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Explicitly, the disjoint union topology can be described as follows. A subset
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The disjoint union of two or more nonempty topological spaces is
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In addition to being continuous, the canonical injections ฯ
664:. It follows from the above universal property that a map
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originates from the fact that the disjoint union is the
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229:} be a family of topological spaces indexed by
546:. Yet another formulation is that a subset
61:. Unsourced material may be challenged and
656:This shows that the disjoint union is the
652:Characteristic property of disjoint unions
337:{\displaystyle \varphi _{i}:X_{i}\to X\,}
333:
125:Learn how and when to remove this message
641:such that the following set of diagrams
421:for which all the canonical injections
14:
906:
803:Preservation of topological properties
460:induced by the canonical injections).
730:. It follows that the injections are
398:{\displaystyle \varphi _{i}(x)=(x,i)}
522:{\displaystyle \varphi _{i}^{-1}(U)}
59:adding citations to reliable sources
26:
743:may be canonically thought of as a
272:{\displaystyle X=\coprod _{i}X_{i}}
177:is a space formed by equipping the
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599:is a topological space, and
441:{\displaystyle \varphi _{i}}
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776:, then the disjoint union
587:The disjoint union space
848:Every disjoint union of
834:Every disjoint union of
820:Every disjoint union of
808:Every disjoint union of
880:, the dual construction
780:is homeomorphic to the
407:disjoint union topology
187:disjoint union topology
732:topological embeddings
696:is continuous for all
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561:its intersection with
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141:and related areas of
728:open and closed maps
625:, then there exists
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554:is open relative to
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55:improve this article
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349:canonical injection
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593:universal property
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413:is defined as the
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175:topological spaces
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893:topological union
888:quotient topology
884:subspace topology
797:discrete topology
772:to a fixed space
452:(i.e.: it is the
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16:(Redirected from
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914:General topology
878:product topology
850:Hausdorff spaces
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70:Find sources:
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40:This article
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864:disconnected
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852:is Hausdorff
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115:October 2009
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53:Please help
41:
812:is discrete
529:is open in
185:called the
143:mathematics
900:References
816:Separation
583:Properties
450:continuous
208:Definition
155:free union
151:direct sum
85:newspapers
658:coproduct
575:for each
538:for each
503:−
494:φ
430:φ
360:φ
328:→
306:φ
251:∐
194:coproduct
192:The name
167:coproduct
42:does not
908:Category
872:See also
795:has the
759:If each
755:Examples
745:subspace
713: :
668: :
633: :
606: :
484:preimage
221: :
159:free sum
787:×
660:in the
643:commute
405:). The
347:be the
282:be the
200:of the
169:) of a
99:scholar
63:removed
48:sources
840:spaces
826:spaces
791:where
294:, let
233:. Let
171:family
145:, the
101:
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212:Let {
165:, or
106:JSTOR
92:books
842:is T
828:is T
726:are
482:its
473:open
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78:news
46:any
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768:is
747:of
700:in
690:o ฯ
678:iff
559:iff
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475:in
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467:of
456:on
417:on
409:on
290:in
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137:In
57:by
910::
799:.
751:.
722:โ
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686:=
672:โ
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637:โ
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103:ยท
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