Limit point

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Definition

Common form

For a Topological space (X,J), xX is a limit point of A if every neighbourhood 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 xClosure(A{x}) (you can read about closure here)


TODO: Prove these are the same


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