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178:. 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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580:, together with the canonical injections, can be characterized by the following
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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
653:. 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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218:} be a family of topological spaces indexed by
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535:. Yet another formulation is that a subset
50:. Unsourced material may be challenged and
645:This shows that the disjoint union is the
641:Characteristic property of disjoint unions
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326:{\displaystyle \varphi _{i}:X_{i}\to X\,}
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114:Learn how and when to remove this message
630:such that the following set of diagrams
410:for which all the canonical injections
792:Preservation of topological properties
449:induced by the canonical injections).
719:. It follows that the injections are
387:{\displaystyle \varphi _{i}(x)=(x,i)}
7:
511:{\displaystyle \varphi _{i}^{-1}(U)}
48:adding citations to reliable sources
732:may be canonically thought of as a
261:{\displaystyle X=\coprod _{i}X_{i}}
166:is a space formed by equipping the
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275:of the underlying sets. For each
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63:"Disjoint union" topology
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588:is a topological space, and
430:{\displaystyle \varphi _{i}}
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765:, then the disjoint union
576:The disjoint union space
837:Every disjoint union of
823:Every disjoint union of
809:Every disjoint union of
797:Every disjoint union of
869:, the dual construction
769:is homeomorphic to the
396:disjoint union topology
176:disjoint union topology
721:topological embeddings
685:is continuous for all
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130:and related areas of
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614:, then there exists
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44:improve this article
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338:canonical injection
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582:universal property
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402:is defined as the
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164:topological spaces
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877:quotient topology
873:subspace topology
786:discrete topology
761:to a fixed space
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59:Find sources:
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29:This article
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853:disconnected
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104:October 2009
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42:Please help
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801:is discrete
518:is open in
174:called the
132:mathematics
889:References
805:Separation
572:Properties
439:continuous
197:Definition
144:free union
140:direct sum
74:newspapers
647:coproduct
564:for each
527:for each
492:−
483:φ
419:φ
349:φ
317:→
295:φ
240:∐
183:coproduct
181:The name
156:coproduct
31:does not
897:Category
861:See also
784:has the
748:If each
744:Examples
734:subspace
702: :
657: :
622: :
595: :
473:preimage
210: :
148:free sum
776:×
649:in the
632:commute
394:). The
336:be the
271:be the
189:of the
158:) of a
88:scholar
52:removed
37:sources
829:spaces
815:spaces
780:where
283:, let
222:. Let
160:family
134:, the
90:
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76:
69:
61:
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201:Let {
154:, or
95:JSTOR
81:books
831:is T
817:is T
715:are
471:its
462:open
437:are
67:news
35:any
33:cite
757:is
736:of
689:in
679:o ฯ
667:iff
548:iff
539:of
464:in
460:is
456:of
445:on
406:on
398:on
279:in
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46:by
899::
788:.
740:.
711:โ
693:.
675:=
661:โ
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626:โ
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568:.
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