Adherent point explained

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, thusx \in \operatorname{Cl} _{X} A if and only if for all open subsets U \subseteq X, if x \in U then U \cap A ≠ \varnothing.

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 S \subseteq X \subseteq Y, 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,

S \subseteq X \subseteq Y and x \in X. Assuming that x \in \operatorname{Cl} _{X} S, let V be a neighborhood of x in Y so that x \in \operatorname{Cl} _{Y} S will follow once it is shown that V \cap S ≠ \varnothing. The set U : = V \cap X is a neighborhood of x in X (by definition of the subspace topology) so that x \in \operatorname{Cl} _{X} S implies that \varnothing ≠ U \cap S. Thus \varnothing ≠ U \cap S = (V \cap X) \cap S \subseteq V \cap S, as desired. For the converse, assume that x \in \operatorname{Cl} _{Y} S and let U be a neighborhood of x in X so that x \in \operatorname{Cl} _{X} S will follow once it is shown that U \cap S ≠ \varnothing. By definition of the subspace topology, there exists a neighborhood V of x in Y such that U = V \cap X. Now x \in \operatorname{Cl} _{Y} S implies that \varnothing ≠ V \cap S. From S \subseteq X it follows that S = X \cap S and so \varnothing ≠ V \cap S = V \cap (X \cap S) = (V \cap X) \cap S = U \cap S, as desired. \blacksquare

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 \left(x _{n\right)} _{n} 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 0 , 0 , \ldots whose limit is 0 , but 0 is not a limit point of S; it is only an adherent point of S.

References

• Adamson, Iain T., A General Topology Workbook, Birkhäuser Boston; 1st edition (November 29, 1995). .
• Apostol, Tom M., Mathematical Analysis, Addison Wesley Longman; second edition (1974).
• Lipschutz, Seymour; Schaum's Outline of General Topology, McGraw-Hill; 1st edition (June 1, 1968). .
• L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, (1970) Holt, Rinehart and Winston, Inc..

Notes and References

1. Steen, p. 5; Lipschutz, p. 69; Adamson, p. 15.