accumulation point
التعريفات والمعاني
== Anglais ==
=== Étymologie ===
Étymologie manquante ou incomplète. Si vous la connaissez, vous pouvez l’ajouter en cliquant ici.
=== Locution nominale ===
accumulation point \Prononciation ?\
(Topologie) Point d’accumulation.
LEMMA 5.2 Let
X
{\displaystyle X}
be a Hausdorff space and
A
{\displaystyle A}
a subset of
X
{\displaystyle X}
. A point
a
∈
X
{\displaystyle a\in X}
is an accumulation point of
A
{\displaystyle A}
if and only if
a
{\displaystyle a}
is a limit point of
A
{\displaystyle A}
. — (Bert Mendelson, Introduction to Topology, Dover Publications, Inc., New York, 1990, 3e édition (1re édition 1975), page 173, ISBN 0-486-66352-3 (OCLC 932232056), § 5.3)
La traduction en français de l’exemple manque. (Ajouter)
A set
A
{\displaystyle A}
has an accumulation point
p
{\displaystyle p}
if for every
ϵ
>
0
{\displaystyle \textstyle \epsilon >0}
there is an
x
∈
A
{\displaystyle \textstyle x\in A}
with
x
≠
p
{\displaystyle \textstyle x\neq p}
and
|
x
−
p
|
<
ϵ
{\displaystyle \textstyle \vert x-p\vert <\epsilon }
. Informally,
p
{\displaystyle p}
is an accumulation point of
A
{\displaystyle A}
if there are points of
A
{\displaystyle A}
that are arbitrarily close to
p
{\displaystyle p}
. Note that the fact that
p
{\displaystyle p}
is an accumulation point of the set
A
{\displaystyle A}
has nothing to do with whether
p
{\displaystyle p}
is actually an element of
A
{\displaystyle A}
. For example, the set
A
=
{
1
n
|
n
∈
N
}
{\displaystyle \textstyle A=\{{\frac {1}{n}}\vert n\in \mathbb {N} \}}
has one accumulation point,
0
{\displaystyle 0}
, because for every
ϵ
>
0
{\displaystyle \textstyle \epsilon >0}
there is an
n
∈
N
{\displaystyle \textstyle n\in \mathbb {N} }
with
1
n
<
ϵ
{\displaystyle \textstyle {\frac {1}{n}}<\epsilon }
. Here the accumulation point
0
{\displaystyle 0}
is not an element of the set
A
{\displaystyle A}
. — (Jonathan M. Kane, Writing Proofs in Analysis, Springer, 2016, page 74)
La traduction en français de l’exemple manque. (Ajouter)
(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
∞
≈
3.570
{\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)
La traduction en français de l’exemple manque. (Ajouter)
==== Synonymes ====
cluster point
==== Antonymes ====
isolated point
==== Hyperonymes ====
limit point
==== Dérivés ====
complete accumulation point
=== Prononciation ===
→ Prononciation audio manquante. (Ajouter un fichier ou en enregistrer un avec Lingua Libre )
=== Voir aussi ===
limit set
accumulation point sur l’encyclopédie Wikipédia (en anglais)
Accumulation point sur Encyclopædia of Mathematics