Ambiguity definitions - accumulationpoint The literature is a bit ambiguous in my point of view. Limit points and accumulation points seems to be the same. I can accept that; that's just two names for the same. But I've seen different definitions of accumulation points.
Definition 1: Let $A\subset X$ be a metric space. A point $a\in A$ is called an accumulation point of $A$ if there exists a sequence $(x_{n})\in A$ such that $x_{n}\in B_{\frac{1}{n}}(a)$, and hence $\lim_{n\rightarrow\infty}x_{n}=a$.
Definition 2: Let $A$ be a metric space. A point $a\in A$ is called an accumulation point for a sequence $(x_{n})$ if for every $\epsilon >0$ there exists some $x_{n}\neq a$ such that $x_{n}\in B_{\epsilon}(a)$.
Definition 3: A point $a$ is called an accumulation point of the set $A$ if every neighborhood of $a$ contains a point $q\neq a$ such that $q\in A$.
So first of all. Do we distinguish accumulation points for sets, from accumulation points for sequences, and are these definitions equivalent ?
 A: The thing is that, depending in the book one is referring, the definitions, as such change. It is just a matter of convention. Basically there are 2 points worth talking about, the limit point and say the accumulation point. 
Define limit point of a set A as: 
Let X be a topological space, then $a\in X$ is a limit point of A if intersection of every open set, containing a, with A is non-empty, that is, given any open set U, such that $a\in U$, there exists a point of A in U.
Define accumulation point of a set A as:
Let X be a topological space, then $a\in X$ is an accumulation point of A if, given an open set U, such that $a\in U$, then there exists a point of A in U, other than a.
The difference is just the last part - other than a.
So your Definition 1 is that of a limit point, while Definition 2 and 3 are that of accumulation point. 
Note that every accumulation point is a limit point. But every limit point need not be an accumulation point. 
For example, take $\Bbb R$ with the discrete topology, that is every subset of $\Bbb R$ is an open set. Consider the singleton set A={0}. 0 is a limit point of A but it is not an accumulation point. Rather there does not exist any accumulation point of A.
As for the sequence, we talk about limit points and accumulation points of a sequence by just considering the set A as the set of all the points of the sequence. And it is interchangeable, in the sense that given a set A and its limit and accumulation points, one can construct a sequence with the same limit or accumulation point. The limit point of a sequence is the point of convergence of the sequence.
Essentially there is but a small difference. And so far for me the use of accumulation point has been very little.
Hope this helps.
