Difference between revisions of "Connected (topology)"

Revision as of 14:09, 13 February 2015 (view source) Alec (Talk | contribs) m ← Older edit		Revision as of 19:47, 14 February 2015 (view source) Alec (Talk | contribs) m Newer edit →
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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}}

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Revision as of 19:47, 14 February 2015

Definition

A topological space

$$(X,\mathcal{J})$$ 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 $$U\cap V=\emptyset$$ where $$U\cup V=X$$

Equivalent definition

We can also say: A topological space

$$(X,\mathcal{J})$$ is connected if and only if the sets $$X,\emptyset$$ are the only two sets that are both open and closed.

Proof

Connected $$\implies$$ only sets both open and closed are $$X,\emptyset$$

Suppose

$$X$$ is connected and there exists a set $$A$$ that is not empty and not all of $$X$$ which is both open and closed. Then as this is closed, $$X-A$$ is open. Thus $$A,X-A$$ is a separation, contradicting that $$X$$ is connected.

Only sets both open and closed are $$X,\emptyset\implies$$ connected

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TODO:

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