Limit Point Definition in Topology For my undergrad real analysis class we are using a book called Principals of Mathematical Analysis and it has a chapter on Topology.  However, I'm pretty confused on what a limit point is. It says "A point p is a limit point of the set E if every neighborhood of p contains a point q $\neq$ p such that q $\in$ E"  But doesn't P only have one neighborhood?  Can anyone offer me an intuitive explanation of what a limit point it? Thanks! 
 A: The neighborhoods of a point $p$ are all the sets $F$ containing an open set $U$ containing $p$. So, no, $p$ has many neighborhoods, uncountably many in most examples. For this definition it's enough to take open neighborhoods: $p$ is a limit point of $E$ if every open set $U$ containing $p$ also contains a point $q\neq p$ in $E$. For further intuition, on the real line you can take the sets $U$ to be just $(p-1/n,p+1/n)$ as $n$ runs from $1$ to $\infty$: then the definition says that there are points of $E$, other than $p$, arbitrarily close to $p$. And that's how you should think about neighborhoods in general. 
A: Points in a topological space can have many neighborhoods, even uncountably many.  If $p$ is a limit point of $E$ then every neighborhood of $p$ contains some point in $E$ that is distinct from $p$ (note $p$ need not even be an element of $E$ to be a limit point). To illustrate this consider the interval $(0,1)$.  Which points in $\mathbb{R}$ have neighborhoods (open intervals) that always contain some point in $(0,1)$?
