accumulation point

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Étymologie

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accumulation point \Prononciation ?\

1. (Topologie) Point d’accumulation.
   • LEMMA 5.2    Let $X$ be a Hausdorff space and $A$ a subset of $X$. A point $a\in X$ is an accumulation point of $A$ if and only if $a$ is a limit point of $A$. — (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 $A$ has an accumulation point $p$ if for every $\textstyle \epsilon >0$ there is an $\textstyle x\in A$ with $\textstyle x\neq p$ and $\textstyle \vert x-p\vert <\epsilon$. Informally, $p$ is an accumulation point of $A$ if there are points of $A$ that are arbitrarily close to $p$. Note that the fact that $p$ is an accumulation point of the set $A$ has nothing to do with whether $p$ is actually an element of $A$. For example, the set $\textstyle A=\{{\frac {1}{n}}\vert n\in \mathbb {N} \}$ has one accumulation point, $0$, because for every $\textstyle \epsilon >0$ there is an $\textstyle n\in \mathbb {N}$ with $\textstyle {\frac {1}{n}}<\epsilon$. Here the accumulation point $0$ is not an element of the set $A$. — (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 ($a_{\infty }\approx 3.570$ 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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Synonymes

• cluster point

Antonymes

• isolated point

Hyperonymes

• limit point

Dérivés

• complete accumulation point

Prononciation

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Voir aussi

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