Limit Point/Examples/Union of Singleton with Open Real Interval

Examples of Limit Points
Let $\R$ be the set of real numbers.

Let $H \subseteq \R$ be the subset of $\R$ defined as:
 * $H = \set 0 \cup \openint 1 2$

Then $0$ is not a limit point of $H$.

Proof
Consider the open $1$- ball $\map {B_1} 0$ of $0$.

Then the only element of $H$ which is in $\map {B_1} 0$ is $0$ itself.

Hence $0$ is not a limit point of $H$ by definition.