A limit point is just a accumulation point whose neighbourhood contains infinitely many elements of the sequence.
Is there any difference between boundary point & limit point? I've read in another question here that all boudary points are limit points, but is the converse true?
Best Answer
Definition of Limit Point: "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."
~from Wikipedia
Definition of Boundary: "Let $S$ be a subset of a topological space $X$. The boundary of $S$ is the set of points $p$ of $X$ such that every neighborhood of $p$ contains at least one point of $S$ and at least one point not of $S$."
~from Wikipedia
So deleted neighborhoods of limit points must contain at least one point in $S$. But (not necessarily deleted) neighborhoods of boundary points must contain at least one point in $S$ AND one point not in $S$.
So they are not the same.
Consider the set $S=\{0\}$ in $\Bbb R$ with the usual topology. $0$ is a boundary point but NOT a limit point of $S$.
Consider the set $S'=[0,1]$ in $\Bbb R$ with the usual topology. $0.5$ is a limit point but NOT a boundary point of $S'$.