(Mathematics - Set Theory and Topology)

Definition of limit point
Let x be a point and S be a subset of a metric space M. Then x is a limit point of S if the closure of S - {x} includes x.

Definition of closure
For any subset A in M, A' is the closure of A if it is the intersection of every closed subset of M that contains A. In other words, A' is the smallest closed set in M that contains A.

The above definitions come from Set Theory and Metric Spaces by Kaplansky © 1972. A different definition of limit point comes from The Advanced Calculus of One Variable by Lick © 1971. Lick's version is as follows:

Definition of limit point
Let S be a set of real numbers. Point x is a limit point of S if there exists infinitely many elements of S in each neighborhood of x.

In Complex Variables and Applications 6th Ed. Brown & Churchill © 1996, the term accumulation point is given a definition similar to Lick's definition of limit point except for the broader complex numbers.

In Eric Weisstein's World of Mathematics, limit point is defined similarly to Kaplansky's definition, except it applies to the broader topological spaces.

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