Limit point

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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Examples

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

Proof using first definition

Is is clear we are talking about the Euclidian metric