Difference between revisions of "Connected (topology)"
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==Equivalent definition== | ==Equivalent definition== | ||
We can also say: A topological space <math>(X,\mathcal{J})</math> is connected if and only if the sets <math>X,\emptyset</math> are the only two sets that are both open and closed. | We can also say: A topological space <math>(X,\mathcal{J})</math> is connected if and only if the sets <math>X,\emptyset</math> are the only two sets that are both open and closed. | ||
| + | {{Begin Theorem}} | ||
| + | Theorem: A topological space <math>(X,\mathcal{J})</math> is connected if and only if the sets <math>X,\emptyset</math> are the only two sets that are both open and closed. | ||
| + | {{Begin Proof}} | ||
| + | '''Connected<math>\implies</math>only sets both open and closed are <math>X,\emptyset</math>''' | ||
| + | :Suppose <math>X</math> is connected and there exists a set <math>A</math> that is not empty and not all of <math>X</math> which is both open and closed. Then as :this is closed, <math>X-A</math> is open. Thus <math>A,X-A</math> is a separation, contradicting that <math>X</math> is connected. | ||
| − | + | '''Only sets both open and closed are <math>X,\emptyset\implies</math>connected''' | |
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{{Todo}} | {{Todo}} | ||
| + | {{End Proof}} | ||
| + | {{End Theorem}} | ||
| + | ==Connected subset== | ||
| + | Let {{M|A}} and {{M|B}} be two [[Subspace topology|topological subspaces]] - they are separated if each is disjoint from the [[Closure, interior and boundary|closure]] of the other (closure in {{M|X}}), that is: | ||
| + | * <math>(B\cap \bar{A})\cup(A\cap\bar{B})=\emptyset</math> | ||
{{Definition|Topology}} | {{Definition|Topology}} | ||
Revision as of 18:21, 19 April 2015
Definition
A topological space [math](X,\mathcal{J})[/math] is connected if there is no separation of [math]X[/math]
Separation
This belongs on this page because a separation is only useful in this definition.
A separation of [math]X[/math] is a pair of two non-empty open sets [math]U,V[/math] where [math]U\cap V=\emptyset[/math] where [math]U\cup V=X[/math]
Equivalent definition
We can also say: A topological space [math](X,\mathcal{J})[/math] is connected if and only if the sets [math]X,\emptyset[/math] are the only two sets that are both open and closed.
Theorem: A topological space [math](X,\mathcal{J})[/math] is connected if and only if the sets [math]X,\emptyset[/math] are the only two sets that are both open and closed.
Connected[math]\implies[/math]only sets both open and closed are [math]X,\emptyset[/math]
- Suppose [math]X[/math] is connected and there exists a set [math]A[/math] that is not empty and not all of [math]X[/math] which is both open and closed. Then as :this is closed, [math]X-A[/math] is open. Thus [math]A,X-A[/math] is a separation, contradicting that [math]X[/math] is connected.
Only sets both open and closed are [math]X,\emptyset\implies[/math]connected
TODO:
Connected subset
Let [ilmath]A[/ilmath] and [ilmath]B[/ilmath] be two topological subspaces - they are separated if each is disjoint from the closure of the other (closure in [ilmath]X[/ilmath]), that is:
- [math](B\cap \bar{A})\cup(A\cap\bar{B})=\emptyset[/math]
