The main difference between an open set and a closed set is a closed set includes its boundary while an open set does not. However, in topology, a closed set is also distinguished (from an open set) as one that includes all its limit points. Does this mean all limit points can only be in the boundary set? What am I missing here?
[Math] Limit points and boundary sets in topology
general-topology
Best Answer
One definition of the boundary $\partial X$ of a set $X$ is that it is the set of all points $p$ so that every open neighborhood of $p$ intersects both $X$ and $X^C$ (the complement of $X$). This means in particular that every point in $\partial X$ is a limit point of $X$, so any closed set must contain its boundary.
However, a set $X$ may have limit points not contained in the boundary: in particular, take $X = \mathbb{R}$. Then, $X$ has no boundary, but every point of $X$ is a limit point.
It is interesting to notice, however, that if a set $Y$ fails to contain a limit point $p$, then this point must lie on the boundary (because $p$ is a limit point of $Y$, every neighborhood of $p$ contains a point of $Y$, and since $p \not\in Y$, every neighborhood of $p$ contains a point not in $Y$).