Difference between revisions of "Connected (topology)"

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(Created page with " ==Definition== A topological space <math>(X,\mathcal{J})</math> is connected if there is no separation of <math>X</math> ===Separation=== This belongs...")
 
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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==
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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.
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===Proof===
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{{Todo}}
  
 
{{Definition|Topology}}
 
{{Definition|Topology}}

Revision as of 14:09, 13 February 2015

Definition

A topological space (X,J) is connected if there is no separation of X

Separation

This belongs on this page because a separation is only useful in this definition.

A separation of X is a pair of two non-empty open sets U,V where UV= where UV=X

Equivalent definition

We can also say: A topological space (X,J) is connected if and only if the sets X, are the only two sets that are both open and closed.

Proof


TODO: