What is 'cluster point' in french?

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The term "cluster point" in French mathematical literature is translated as "valeur d'adhérence." This concept is related to adherent points, which are defined such that every epsilon-ball around the point contains at least one point from the set. The discussion highlights that while every cluster point is adherent, not every adherent point qualifies as a cluster point. Additionally, "point d'accumulation" is mentioned as a related term, but it faces similar definitional challenges. The conversation emphasizes the nuances in definitions within the context of metric spaces and sequences.
quasar987
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I'm not talking about how you would personally translate it.. I'm asking what is 'cluster point' in the french mathematical literature (textbooks).
 
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I believe it's "aaleur d'adhérence". All I did is wikipedia "Cluster Point" and then I clicked "français" on the left.
 
In english, an adherent point for a set E is a point x for which "for each epsilon>0, the epsilon-ball centered on x contains at least one point of E". This implies in particular that every point of E is adherent. So even if we consider the image of a sequence, {x1,x2,...}, every cluster point is adherent but not every adherent point is a cluster point.

So if cluster point = valeur d'adhérence, I wonder what "adhenrent point" is in french!
 
Point d'accumulation
 
'Point d'accumulation' runs into the same "problem" as 'valeur d'adhérence' because as defined for a set, an accumulation point of a subset A of a metric space M is a point a of M for which for all e>0, (B_{\epsilon}(a)\backslash \{a\})\cap A\neq \emptyset (i.e. every open ball centered on a contains points of A other than a).

So even if we apply this concept to the image of the sequence, every accumulation point is a cluster point but not every cluster point is an accumulation point. For instance consider the constant sequence 1,1,1,... 1 is a cluster point but the set of all accumulation points of the image, {1}, is void.
 
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Have you considered peeking at a math french/english dictionary at some library?
 
This thing exists?!
 
How do you define "cluster point"?
 
If {x_n} is a sequence in a metric space M, then a clsuter point of {x_n} is a point a of M such that there is a subsequence of {x_n} converging to a.
 
  • #10
Soit E un espace topologique. A une partie non vide de E, et aE. On dit que le point a est un point d'accumulation de A s'il est adhérent à A sans être isolé dans A. Autrement dit, a est un point d'adhérence de A-{a}.

Exemple : A=[0,1[, 1 est un point d'accumulation.

Si E est un espace métrique (en particulier un espace vectoriel normé), on montre facilement que les points suivants sont équivalents :

a est point d'accumulation de A
il existe une suite injective de points de A convergeant vers a.
tout voisinage de a contient une infinité de points de A.
Ceci montre en particulier que l'ensemble des points d'accumulation de A est un fermé.

Taken from DicoMath (french website)
 
  • #11
I don't see how this helps but nice site, thx.
 
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