Difference between revisions of "Disconnected (topology)/Definition"
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{{Requires references|grade=A|msg=Mendelson and Lee's topological manifolds have it covered, I think Munkres is where I got "separation" from}} | {{Requires references|grade=A|msg=Mendelson and Lee's topological manifolds have it covered, I think Munkres is where I got "separation" from}} | ||
==Definition== | ==Definition== | ||
| − | </noinclude>A [[topological space]], {{Top.|X|J}}, is said to be ''disconnected'' if: | + | </noinclude>A [[topological space]], {{Top.|X|J}}, is said to be ''disconnected'' if{{rITTMJML}}: |
* {{M|1=\exists U,V\in\mathcal{J}[U\ne\emptyset\wedge V\neq\emptyset\wedge V\cap U=\emptyset\wedge U\cup V=X]}}, in words "''if there exists a pair of [[disjoint]] and ''[[non-empty]]'' [[open sets]], {{M|U}} and {{M|V}}, such that their [[union]] is {{M|X}}''" | * {{M|1=\exists U,V\in\mathcal{J}[U\ne\emptyset\wedge V\neq\emptyset\wedge V\cap U=\emptyset\wedge U\cup V=X]}}, in words "''if there exists a pair of [[disjoint]] and ''[[non-empty]]'' [[open sets]], {{M|U}} and {{M|V}}, such that their [[union]] is {{M|X}}''" | ||
| − | In this case, {{M|U}} and {{M|V}} are said to ''disconnect {{M|X}}'' and are sometimes called a ''separation of {{M|X}}''.<noinclude> | + | In this case, {{M|U}} and {{M|V}} are said to ''disconnect {{M|X}}''<ref name="ITTMJML"/> and are sometimes called a ''separation of {{M|X}}''.<noinclude> |
==References== | ==References== | ||
Latest revision as of 22:15, 30 September 2016
Grade: A
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Mendelson and Lee's topological manifolds have it covered, I think Munkres is where I got "separation" from
Definition
A topological space, [ilmath](X,\mathcal{ J })[/ilmath], is said to be disconnected if[1]:
- [ilmath]\exists U,V\in\mathcal{J}[U\ne\emptyset\wedge V\neq\emptyset\wedge V\cap U=\emptyset\wedge U\cup V=X][/ilmath], in words "if there exists a pair of disjoint and non-empty open sets, [ilmath]U[/ilmath] and [ilmath]V[/ilmath], such that their union is [ilmath]X[/ilmath]"
In this case, [ilmath]U[/ilmath] and [ilmath]V[/ilmath] are said to disconnect [ilmath]X[/ilmath][1] and are sometimes called a separation of [ilmath]X[/ilmath].
