Difference between revisions of "Limit point"

Revision as of 21:15, 12 February 2015 (view source) Alec (Talk | contribs) (Created page with "{{Definition|Topology|Metric Space}} ==Definition== ===Common form=== For a Topological space <math>(X,\mathcal{J})</math>, <math>x\in X</math> is a limit point of <math>...")		Revision as of 00:35, 13 February 2015 (view source) Alec (Talk | contribs) m Newer edit →
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	<math>x</math> is a limit point of <math>A</math> if <math>x\in\text{Closure}(A-\{x\})</math> (you can read about [[Closure|closure here]])		<math>x</math> is a limit point of <math>A</math> if <math>x\in\text{Closure}(A-\{x\})</math> (you can read about [[Closure|closure here]])
	{{Todo|Prove these are the same}}		{{Todo|Prove these are the same}}
		+	==Other names==
		+	* Accumilation point
	==Examples==		==Examples==
	===<math>0</math> is a limit point of <math>(0,1)</math>===		===<math>0</math> is a limit point of <math>(0,1)</math>===

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Revision as of 00:35, 13 February 2015

Definition

Common form

For a Topological space

$$(X,\mathcal{J})$$ , $$x\in X$$ is a limit point of $$A$$ if every neighborhood of $$x$$ has a non-empty intersection with $$A$$ that contains some point other than $$x$$ itself.

Equivalent form

$$x$$ is a limit point of $$A$$ if $$x\in\text{Closure}(A-\{x\})$$ (you can read about closure here)

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TODO: Prove these are the same

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Other names

• Accumilation point

Examples

$$0$$ is a limit point of $$(0,1)$$

Proof using first definition

Is is clear we are talking about the Euclidian metric

Proof using second definition