Definition. A point $x$ is a limit point of a set A if every $\varepsilon$-neighborhood of $x$ intersects the set A in some point other than $x$.
I understand the definition in that $x$ is our limit point for the set A. What I don't understand is where it says "intersects" the set A in some point other than $x$. Can some one explain what is meant by "intersect"?
For example in $\Bbb R$ the subset $\Bbb N$ has'nt any limit point:
If $n \in \Bbb N$ then for example $]n-1/2,n+1/2[$ intersects $\Bbb N$ only in $\{n\}$.
$A=\left\{0,1,\frac 12, \frac 13, ....\right\}$ has a unique limite point : that is $0$.
because for all $\varepsilon > 0$, we have : $(]-\varepsilon,\varepsilon[)\cap A =\{\frac 1p,p>\frac 1{\varepsilon}\} $ but for $n \in \Bbb N, n \neq 0$ and $\epsilon=\frac{1}{2n(n+1)}$ we have $A \cap (]-\varepsilon,\varepsilon[)=\{\frac 1n \}$