Difference between revisions of "Connected (topology)"
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A separation of <math>X</math> is a pair of two non-empty [[Open set|open sets]] <math>U,V</math> where <math>U\cap V=\emptyset</math> where <math>U\cup V=X</math> | A separation of <math>X</math> is a pair of two non-empty [[Open set|open sets]] <math>U,V</math> where <math>U\cap V=\emptyset</math> where <math>U\cup V=X</math> | ||
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| + | ==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=== | ||
| + | {{Todo}} | ||
{{Definition|Topology}} | {{Definition|Topology}} | ||
Revision as of 14:09, 13 February 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.
Proof
TODO:
