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accumulation point

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accumulation point

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Accumulation point

accumulation point

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accumulation point

出典:『Wiktionary』 (2024/08/08 18:22 UTC 版)

名詞

accumulation point (plural accumulation points)

(topology, "of" a subset of a topological space) 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.
Synonyms: cluster point, limit point
- 1975, Bert Mendelson, Introduction to Topology, 3rd edition, New York: Dover Publications, Inc., published 1990, →ISBN, →OCLC, §5.3, page 173:
- 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.
(mathematical analysis, "of" a sequence) Given a sequence si, a point x whose every neighborhood contains at least one element of the sequence distinct from x.
Synonyms: cluster point, limit point
(systems theory, dynamical systems, chaos theory) For certain maps, a point beyond which periodic orbits give way to chaotic ones.
- 1995, Milos Marek, Igor Schreiber, Chaotic Behaviour of Deterministic Dissipative Systems, Cambridge University Press, page 77:
  The chaotic set (not necessarily attracting) is formed after the first accumulation point ( a ∞ ≈ 3.570 {\displaystyle a_{\infty }\approx 3.570} $a_{\infty }\approx 3.570$ for the logistic mapping) is reached. In the chaotic region of the logistic map the periodicity re-emerges in periodic windows which are bounded by the accumulation point from the right and by the saddle-node bifurcation from the left. A reverse bifurcation sequence occurs above the accumulation point.

使用する際の注意点

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.
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.

派生語

complete accumulation point

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