Accumulation point

English

Noun

1. Template:Lb Given a subset S of a topological space X, a point x whose every neighborhood contains at least one point distinct from x that belongs to S.

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   • 2008, Brian S. Thomson, Andrew M. Bruckner, Judith B. Bruckner, Elementary Real Analysis, Volume 1, Thomson-Bruckner (ClassicalRealAnalysis.com), 2nd Edition, page 153,

     Definition 4.9 (Closed): The set E is said to be closed provided that every accumulation point of E belongs to the set E.

     Thus a set E is not closed if there is some accumulation point of E that does not belong to E. In particular, a set with no accumulation points would have to be closed since there is no point that needs to be checked.

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2. Template:Lb Given a sequence s_i, a point x whose every neighborhood contains at least one element of the sequence distinct from x.

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3. Template:Lb For certain maps, a point beyond which periodic orbits give way to chaotic ones.
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Usage notes

• If X is a T₁ space (a broad class that includes Hausdorff spaces and metric spaces), then the set of points in S in each neighborhood of an accumulation point x is at least countably infinite.
  • If each neighborhood's intersection with S is uncountably infinite, the term Template:M can be used. Terms such as ${\mathrm{\aleph }}_0$-accumulation point (or $\omega$-accumulation point) and ${\mathrm{\aleph }}_1$-accumulation point may also be used.
  • The term Template:M may be used if the cardinality of the set of points in any given neighborhood of x that are also in S is equal to the cardinality of S.
• The sequence case can be regarded as a particular instance of the topological definition. For a sequence of real numbers, for instance, the topological space is the real number line (equipped with an order topology provided by the absolute value metric), of which the sequence is subset. If the sequence has a limit, it must be an accumulation point. (But note that a sequence may have more than one accumulation point.)
  • Consequently, both cases can be explained and discussed in similar mathematical language.

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See also

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Further reading

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