(

*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.