In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an adherent point (also closure point or point of closure) is a slight generalization of the idea of a limit point

Limit point

In mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S...

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If A is a subset of a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 X, then a point x in X is an adherent point of A if every open set

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

 containing x contains at least one point of . A point x is an adherent point for A if and only if x is in the closure

Closure (topology)

In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...

 of A.

This definition is more general than that of a limit point

Limit point

In mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S...

, in that for a limit point it is required that every open set containing contains at least one point of A different from x. Thus every limit point is an adherent point, but the converse fails. 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

Isolated point

In topology, a branch of mathematics, a point x of a set S is called an isolated point of S, if there exists a neighborhood of x not containing other points of S.In particular, in a Euclidean space ,...

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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 subset of a metric space X which is bounded above, then sup S is adherent to S.
• For any subset S of a metric space M, S contains all of its adherent points if, and only if, S is closed in M.
• In the interval (a, b], a is an adherent point that is not in the interval.