Limit point, Accumulation point, and Condensation point of a set

Posted 2020-10-12 Mathematics & Control Theo

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.