TITLE: What is a limit point
QUESTION [32 upvotes]: Wikipedia seems to describe the topic with extreme complexity for me. 

In mathematics, a limit point of a set $S$ in a topological space $X$ is a
  point $x$ (which is in $X$, but not necessarily in $S$) that can be
  "approximated" by points of $S$ in the sense that every neighbourhood of
  $x$ with respect to the topology on $X$ also contains a point of $S$ other
  than $x$ itself. Note that $x$ does not have to be an element of $S$.

I don't understand the relationship between $S$ and $X$ and what it means for $x$ to be a limit point.

REPLY [5 votes]: A point $a$ is said to be a limit point of a set $S$ if there are points in $S$ other than $a$ that are arbitrarily close to $a$ but never become equal to $a$. For example $1$ is a limit point of the intervals $[0,1]$ and $[0,2]$ because $\{0.9,0.99,0.999,0.999 \dots\}$ is a sequence of points in those intervals that approaches $1$ but never become equal to $1$.
Source: Limits, Continuity and Differentiability - David Levermore