TITLE: limit point of the set of integers
QUESTION [6 upvotes]: What are the limit point of the set of integers? I know that the closure of this set is the set of integers, so is this set of limit points empty?

REPLY [6 votes]: The confusion here is that the definition of a limit point $x$ in a set $X$ requires that every neighborhood of $x$ contains a point of $X$ different from $x$. A limit point is also known as an accumulation point.
http://en.wikipedia.org/wiki/Limit_point
This is in contrast to the definition of an adherent point, also known as a contact point, which is a point whose every neighborhood intersects $X$. 
http://en.wikipedia.org/wiki/Adherent_point
If a set is closed, then every one of its points are adherent points; but not necessarily limit points. For example the set $[0, 1] \cup \{2\}$ is a closed set in $\mathbb{R}$. Every point is an adherent point, but $2$ is not a limit point.
This terminology a common point of confusion. 
To answer the original question, the integers have no limit points in the reals, since all integers are isolated; that is, each integer has a neighborhood that does not contain any other integers.