Limit Points and Closure

Limit Point & Isolated Point

Let be a metric space and . is called a limit point of if every open ball contains a point of other than . Conversely, is called an isolated point of if there is an open ball that contains no elements of other than .

e.g.

• Consider in with Euclidean metric. Then is an isolated point of , every point in is a limit point of and the closure of is .
• Let be a normed vector space and . Then for any we have

Closure

The closure of is defined as

Theorem

Let be a metric space and . If is a limit point of , then every open ball contains infinitely many points of .

Proof

Theorem

if and only if every contains a point of .

Proof implies or is a limit point of . Thus every contains a point of . Suppose every contains a point of . If , then . If , then by the given condition, every contains a point of other than . Thus is a limit point of and hence .

Theorem

Let be a metric space and , then

Proof we haveTherefore . Consequently, This and imply . Since , we must have and thus .