Confusing definition of limit points I found this definition on wikipedia.

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.

But doesn't this just mean it could be pretty much any point? Not necessarily anywhere near a boundary/limit of S? Just a random point in S.
For example pick the point, 2, in S (the standard topology of R between (1,3)). Every neighbourhood of x will contain a point in S that is not x.
What am I missing, this notion of a limit point seems pointless..?

The word limit in the definition of limit point its not related to the concept of topological boundary. It is related to the fact that it is the limit of a sequence of points in $S$ different from $x$.
Also, not every point of $S$ is a limit point. Isolated points like $2$ in $(0,1)\cup\{2\}$ are not limit points.
