limit point (metric)
A limit point of a set is a point you can sneak up on using other points of the set, getting as close as you please without ever needing the point itself. There is always company nearby — the set clusters around such a point.
Formally, x is a limit point of A if every open ball around x contains a point of A different from x. Equivalently, there is a sequence of points of A, all distinct from x, converging to x. The point x may or may not belong to A; what matters is that A accumulates there. A set is closed exactly when it contains all of its limit points.
Limit points must be distinguished from isolated points. A point of A is isolated if some small ball around it contains no other point of A — it stands alone. The closure of A is precisely A together with its limit points, which is why limit points are the missing pieces that closure restores.
For A = {1/n : n = 1, 2, 3, ...}, the point 0 is a limit point, since 1/n -> 0 and 0 is not in A. Every point 1/n of A, by contrast, is isolated: it has a small gap to its neighbors.
0 is approached by the set but absent from it — a limit point.