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 neighbourhood of x (or equivalently, every open neighborhood of x) contains at least one point of A. A point x \in X is an adherent point for A if and only if x is in the closure of A, thus This definition differs from that of a limit point of a set, in that for a limit point it is required that every neighborhood of x contains at least one point of A 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 and sufficient conditions

If S is a non-empty subset of \R which is bounded above, then the supremum \sup S is adherent to S. In the interval (a, b], a is an adherent point that is not in the interval, with usual topology of \R. A subset S of a metric space M contains all of its adherent points if and only if S is (sequentially) closed in M.

Adherent points and subspaces

Suppose x \in X and where X is a topological subspace of Y (that is, X is endowed with the subspace topology induced on it by Y). Then x is an adherent point of S in X if and only if x is an adherent point of S in Y. By assumption, and x \in X. Assuming that let V be a neighborhood of x in Y so that will follow once it is shown that The set is a neighborhood of x in X (by definition of the subspace topology) so that implies that Thus as desired. For the converse, assume that and let U be a neighborhood of x in X so that will follow once it is shown that By definition of the subspace topology, there exists a neighborhood V of x in Y such that Now implies that From it follows that and so as desired. Consequently, x is an adherent point of S in X if and only if this is true of x in every (or alternatively, in some) topological superspace of X.

Adherent points and sequences

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 be such a sequence and let x be its limit. Then by definition of limit, for all neighbourhoods U of x there exists n \in \N such that x_n \in U for all n \geq N. In particular, x_N \in U and also x_N \in S, 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 whose limit is 0, but 0 is not a limit point of S; it is only an adherent point of S.

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