In mathematics, an adherent point (also closure point or point of closure or contact point) of a subset of a topological space such that every neighbourhood (or equivalently, every open neighborhood is in the closure thus

This definition differs from that of a limit point of a set (or both). An adherent point which is not a limit point is an isolated point.

including the boundary.

Examples and sufficient conditions

is a non-empty which is bounded above, then the supremum In the interval is an adherent point that is not in the interval, with usual topology

of a metric space is (sequentially) closed

Adherent points and subspaces

is a topological subspace is endowed with the subspace topology

(by definition of the subspace topology By definition of the subspace topology

Adherent points and sequences

is a subset of a topological space then the limit be its limit. Then by definition of limit, for all neighbourhoods

See also

Notes

Citations

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..