Adherent point

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In mathematics, an adherent point (also closure point or point of closure or contact point)[1] of a subset of a topological space is a point in such that every neighbourhood of (or equivalently, every open neighborhood of ) contains at least one point of A point is an adherent point for if and only if is in the closure of thus

if and only if for all open subsets if

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 contains at least one point of different from Thus every limit point is an adherent point, but the converse is not true. An adherent point of is either a limit point of or an element of (or both). An adherent point which is not a limit point is an isolated point.

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

Examples and sufficient conditions

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

A subset   of a metric space   contains all of its adherent points if and only if   is (sequentially) closed in  

Adherent points and subspaces

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Suppose   and   where   is a topological subspace of   (that is,   is endowed with the subspace topology induced on it by  ). Then   is an adherent point of   in   if and only if   is an adherent point of   in  

Proof

By assumption,   and   Assuming that   let   be a neighborhood of   in   so that   will follow once it is shown that   The set   is a neighborhood of   in   (by definition of the subspace topology) so that   implies that   Thus   as desired. For the converse, assume that   and let   be a neighborhood of   in   so that   will follow once it is shown that   By definition of the subspace topology, there exists a neighborhood   of   in   such that   Now   implies that   From   it follows that   and so   as desired.  

Consequently,   is an adherent point of   in   if and only if this is true of   in every (or alternatively, in some) topological superspace of  

Adherent points and sequences

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If   is a subset of a topological space then the limit of a convergent sequence in   does not necessarily belong to   however it is always an adherent point of   Let   be such a sequence and let   be its limit. Then by definition of limit, for all neighbourhoods   of   there exists   such that   for all   In particular,   and also   so   is an adherent point of   In contrast to the previous example, the limit of a convergent sequence in   is not necessarily a limit point of  ; for example consider   as a subset of   Then the only sequence in   is the constant sequence   whose limit is   but   is not a limit point of   it is only an adherent point of  

See also

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Notes

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Citations

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

References

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  • Adamson, Iain T., A General Topology Workbook, Birkhäuser Boston; 1st edition (November 29, 1995). ISBN 978-0-8176-3844-3.
  • Apostol, Tom M., Mathematical Analysis, Addison Wesley Longman; second edition (1974). ISBN 0-201-00288-4
  • Lipschutz, Seymour; Schaum's Outline of General Topology, McGraw-Hill; 1st edition (June 1, 1968). ISBN 0-07-037988-2.
  • L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, (1970) Holt, Rinehart and Winston, Inc..
  • This article incorporates material from Adherent point on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.