英和和英

名詞

accumulation point (plural accumulation points)

1. (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:

     LEMMA 5.2    Let X be a Hausdorff space and A a subset of X. A point ${\displaystyle a\in X}$ is an accumulation point of A if and only if a is a limit point of A.

   • 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.

   • 2016, Jonathan M. Kane, Writing Proofs in Analysis, Springer, page 74:

     A set ${\displaystyle A}$ has an accumulation point ${\displaystyle p}$ if for every ${\displaystyle \textstyle \epsilon >0}$ there is an ${\displaystyle \textstyle x\in A}$ with ${\displaystyle \textstyle x\neq p}$ and ${\displaystyle \textstyle \vert x-p\vert <\epsilon }$. Informally, ${\displaystyle p}$ is an accumulation point of ${\displaystyle A}$ if there are points of ${\displaystyle A}$ that are arbitrarily close to ${\displaystyle p}$. Note that the fact that ${\displaystyle p}$ is an accumulation point of the set ${\displaystyle A}$ has nothing to do with whether ${\displaystyle p}$ is actually an element of ${\displaystyle A}$. For example, the set ${\displaystyle \textstyle A=\{{\frac {1}{n}}\vert n\in \mathbb {N} \}}$ has one accumulation point, ${\displaystyle 0}$, because for every ${\displaystyle \textstyle \epsilon >0}$ there is an ${\displaystyle \textstyle n\in \mathbb {N} }$ with ${\displaystyle \textstyle {\frac {1}{n}}<\epsilon }$. Here the accumulation point ${\displaystyle 0}$ is not an element of the set ${\displaystyle A}$.

2. (mathematical analysis, "of" a sequence) Given a sequence s_i, a point x whose every neighborhood contains at least one element of the sequence distinct from x.

   Synonyms: cluster point, limit point

3. (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 ( ${\displaystyle a_{\infty }\approx 3.570}$ 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.