Difference between revisions of "Connected (topology)"
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===Proof=== | ===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. | ||
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| + | ====Only sets both open and closed are <math>X,\emptyset\implies</math>connected==== | ||
{{Todo}} | {{Todo}} | ||
{{Definition|Topology}} | {{Definition|Topology}} | ||
Revision as of 19:47, 14 February 2015
Contents
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.
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
TODO:
