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

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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 $H$.


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.