Difference between cluster points and limit points? I have a weird doubt about these terms "cluster points" and "limit points". 
I am just calling it weird because I don't know where I am going wrong as I have seen somewhere that cluster points and limit points are one and the same, and somewhere else that they have very slight difference in their meaning.
The set of all limit points of $\mathbb{Z}$ is empty. What about the set of cluster points of $\mathbb{Z}$? Please clear my doubt of their definitions, thanks in advance.
 A: I might be utterly wrong so take this with a grain of salt.
A limit point of A is a point in which every neighborhood has at least one point other than itself of A.
A cluster point of A is a point in which every neighborhood has an infinite number of points of A.
In a metric space these are the equivalent.  (For $a_0$ in the neighborhood of x, find the neighborhood of x with radius $d(a_0, x)/2$ and $a_0$ is not in it so another $a_1$ must be in it.  Inductively repeat.  Thus all limit points are cluster points. [If you accept countable axiom of choice.])
Metric spaces have what is called the T1 axiom which is, apparently (I googled this), that if $a \ne b$ then there exists neighborhoods around each which don't contain the other.  (I interpret this to mean that in a metric space $d(a, b)> 0$ so the neighborhoods of radius $d(a, b)/2$ around each point will not have either point in the other neighborhoods.)  A non-metric space without the T1 axiom  might have two $a \ne b$ such that $b$ is in every neighborhood of $a$!  (Clearly impossible in metric spaces.) Thus if $b \in B$ then $a$ is a limit point of B because $b$ is in every neighborhood.  But $b$ could be the only point in a neighborhood.  Thus $a$ isn't necessarily a cluster point of B.
Am I right?  If I'm not, ignore me.
A: If you are talking about numeric sequences, a point is a cluster point of a sequence if it a limit of a subsequence of that sequence. If every subsequence has the same limit then that point is a limit point for the sequence.  The sequence 1, 1/2, 1/4, 3/4, 1/5, 4/5, 1/6, 5/6, ...  has no limit but has both 0 and 1 as cluster points.  But if you are talking about general sets in a metric space, limit points and cluster points are the same thing- every open neighborhood of the point contains at least one other point of the set.
In both senses, the set of all integers, with the discrete topology, has no limit points and no cluster points.
