Limit point

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)

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TODO: Prove these are the same

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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