accumulation point

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accumulation point \Prononciation ?\

1. (Topologie) Point d’accumulation.
   • LEMMA 5.2    Let ${\displaystyle X}$ be a Hausdorff space and ${\displaystyle A}$ a subset of ${\displaystyle X}$. A point ${\displaystyle a\in X}$ is an accumulation point of ${\displaystyle A}$ if and only if ${\displaystyle a}$ is a limit point of ${\displaystyle A}$. — (Bert Mendelson, Introduction to Topology, Dover Publications, Inc., New York, 1990, 3^e édition (1^re édition 1975), page 173, ISBN 0-486-66352-3 (OCLC 932232056), § 5.3)

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   • 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 ${\displaystyle A}$. — (Jonathan M. Kane, Writing Proofs in Analysis, Springer, 2016, page 74)

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2. (Dynamique des systèmes) Point au-delà duquel les orbites périodiques deviennent chaotiques.
   • 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. — (Milos Marek, Igor Schreiber, Chaotic Behaviour of Deterministic Dissipative Systems, Cambridge University Press, 1995, page 77)

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• cluster point

• isolated point

• limit point

• complete accumulation point

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• limit set
• accumulation point sur l’encyclopédie Wikipédia (en anglais) 
• Accumulation point sur Encyclopædia of Mathematics