In my text book, the definition of a limit point is as follows:

If X is a metric space, a point x of X is a limit point of the subset A of X if every epsilon-neigborhood of x intersects A in at least one point different from x. An equivalent condition is to require that every neigborhood of x contain infinitely many points of A.

Does that mean that, for example, all the points in the interior of an open circle are limit points of that circle?

In a more specific example,

let A={ } where A is a subset of

let Lim(A) be the set of all limit points of A (I am assumingthis is what Lim(A) means btw)

is Lim(A)={ } where Lim(A) is a subset of ?

but then the closure of A, Cl(A)=Lim(A) which makes me question my understanding of the definition of limit points...or is that right?

Intuitively, I think of Lim(A)={0} union { + =1} which is actually Bd(A)