The following is Theorem 28.1. from Munkres' Topology:
It says that for each $a \in A$ we can choose a neighborhood $U_a$ of $a$ such that $U_a$ intersects $A$ in the point $a$ alone.
Is that an assumption or a result of characteristics that $A$ have? And why that assumption/conclusion is correct?
A counterexample for that is when $A=[a,b]$. $A$ contains all its limit points. We can also determine a set $X$ to be compact and to contain $A$. Any neighborhood of any point in $A$ is open interval meaning that it intersects $A$ is infinitely many points other than the point itself.
This is using the assumption that $A$ has no limit points (not merely the statement that $A$ contains all its limit points). Given any $a\in A$, we know $a$ is not a limit point of $A$. By definition of "limit point", this means there is an open neighborhood $U_a$ such that $U_a\cap A$ contains no point of $A$ except possibly $a$. Since $a\in U_a$ and $a\in A$, this means $U_a\cap A=\{a\}$.