Limit points of $(2,3)$. From this question: Constructing a set with exactly three limit points.
To answer this question  

Construct a bounded set of real numbers with exactly three limit points.

But why $(1,2)\bigcup (2,3)$ can't be answer?
From my understand to limit point, $(2,3)$ only have two limit point $2$ and $3$. Any $x$ not be $2$ or $3$, can't be $(2,3)$'s limit point. Am I right?

No.
If you understand why $(2,3)$ is an open set then,
Lemma: Let $X$ be an open set in $\mathbb{R}^k$ , if $p\in X$ then $p$ is a limit point of $X$. (Hint: every neighborhood of $p$ contains an element other than $p$)
Thus every point in $(2,3)$ is a limit point. Also we have two additional limit points, namely $2$ and $3$.
