What's the relationship between interior/exterior/boundary point and limit point? I'm learning real analysis.
I found that there are two classification of points: interior/exterior/boundary point and limit point.
What's the relationship between interior/exterior/boundary point and limit point ?
 A: As an exercise (which should simultaneously answer your questions), prove the following statements:
An interior point cannot be an exterior point.
An exterior point cannot be an interior point.
A boundary point is neither an interior point nor an exterior point.
An exterior point is not a limit point.
An interior point can be a limit point.
Let $S$ be a set. Every boundary point of $S$ is a limit point of $S$ and its complement. (This statement is false if you define a limit point of $S$ to be a point $p$ so that every neighborhood of $p$ contains some $x\in S$, $x\neq p$. But if you allow $x = p$ in the definition then the statement is true.)
These are all trivial, some may be very trivial depending on what the definitions of these terms are for you.
A: From a topological perspective you can say the following: (Assuming $S$ is the subset of a topological space $X$ relative to which you want to define interior/exterior/limit and $S^c$ denotes $S$'s complement)


*

*To every interior point $i$ there is an open subset of $S$ containing $i$.

*To every exterior point $e$ there is an open subset of $S^c$ containing $e$.

*If an open subset of $S$ contains a boundary point $b$ then that subset
contains both points of $S$ and of $S^c$. (Boundary points can be in
either $S$ or $S^c$)

*Neighborhoods of limit points $l$ always contain at least one point
that is in $S$ and is not $l$ itself.


The interior, the boundary and the exterior are a (disjunct) partition of $X$.
Note the difference between the boundary and the limit points: Isolated points (points in $S$ for which holds: there exists an open subset of $X$ that contain the point as sole member of $S$) are in $S$, in the boundary of $S$ but are not limit points of $S$. The Venn-Diagram here shows this nicely.
