The three notions mentioned above should be clearly distinguished.
If $A$ is a subset of a topological space $X$ and $x$ is a point of $X$, then $x$ is an accumulation point of $A$ if and only if every neighbourhood of $x$ intersects $A\backslash \{x\}$.
It is a condensation point of $A$ if and only if every neighbourhood of it contains uncountably many points of $A$.
The term limit point is slightly ambiguous. One might call $x$ a limit point of $A$ if every neighbourhood of $x$ contains infinitely many points of $A$, but this is not standard.
Wikipedia: Let ${S}$ be a subset of a topological space $X$. A point $x$ in $X$ is a limit point of $S$ if every neighbourhood of $x$ contains at least one point of $S$ different from $x$ itself.
Limit point, Accumulation point, and Condensation point of a set
原文：http://www.cnblogs.com/aujun/p/5026285.html