In mathematics, an adherent point (also closure point or point of closure or contact point) of a subset A of a topological space X, is a point x in X such that every open set containing x contains at least one point of A. A point x is an adherent point for A if and only if x is in the closure of A.

This definition differs from that of a limit point, in that for a limit point it is required that every open set containing x contains at least one point of A different from x. Thus every limit point is an adherent point, but the converse is not true. An adherent point of A is either a limit point of A or an element of A (or both). An adherent point which is not a limit point is an isolated point.

Intuitively, having an open set A defined as the area within (but not including) some boundary, the adherent points of A are those of A including the boundary.

## Examples

• If S is a non-empty subset of R which is bounded above, then supS is adherent to S.
• A subset S of a metric space M contains all of its adherent points if, and only if, S is (sequentially) closed in M.
• In the interval (a, b], a is an adherent point that is not in the interval, with usual topology of R.
• If S is a subset of a topological space then the limit of a convergent sequence in S does not necessarily belong to S, however it is always an adherent point of S. Let (xn)nN be such a sequence and let x be its limit. Then by definition, for all open neighbourhoods U of x there exists NN such that xnU for all nN. In particular, xNU and also xNS, so x is an adherent point of S.
• In contrast to the previous example, the limit of a convergent sequence in S is not necessarily a limit point of S; for example consider S = {0} as a subset of R. Then the only sequence in S is the constant sequence (0) whose limit is 0, but 0 is not a limit point of S; it is only an adherent point of S.