Difference between revisions of "Limit point"
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For a [[Topological space]] <math>(X,\mathcal{J})</math>, <math>x\in X</math> is a limit point of <math>A</math> if every [[Neighborhood|neighborhood]] of <math>x</math> has a non-empty [[Intersect|intersection]] with <math>A</math> that contains some point other than <math>x</math> itself. | For a [[Topological space]] <math>(X,\mathcal{J})</math>, <math>x\in X</math> is a limit point of <math>A</math> if every [[Neighborhood|neighborhood]] of <math>x</math> has a non-empty [[Intersect|intersection]] with <math>A</math> that contains some point other than <math>x</math> itself. | ||
===Equivalent form=== | ===Equivalent form=== | ||
| − | <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, Interior and boundary#Closure|closure here]]) |
{{Todo|Prove these are the same}} | {{Todo|Prove these are the same}} | ||
==Other names== | ==Other names== | ||
Revision as of 00:26, 17 February 2015
Contents
Definition
Common form
For a Topological space [math](X,\mathcal{J})[/math], [math]x\in X[/math] is a limit point of [math]A[/math] if every neighborhood of [math]x[/math] has a non-empty intersection with [math]A[/math] that contains some point other than [math]x[/math] itself.
Equivalent form
[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 here)
TODO: Prove these are the same
Other names
- Accumilation point
Examples
[math]0[/math] is a limit point of [math](0,1)[/math]
Proof using first definition
Is is clear we are talking about the Euclidian metric
