# What is 'cluster point' in french?

Homework Helper
Gold Member
I'm not talking about how you would personally translate it.. I'm asking what is 'cluster point' in the french mathematical literature (textbooks).

I believe it's "aaleur d'adhérence". All I did is wikipedia "Cluster Point" and then I clicked "français" on the left.

Homework Helper
Gold Member
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

Homework Helper
Gold Member
'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.

Last edited:
Have you considered peeking at a math french/english dictionary at some library?

Homework Helper
Gold Member
This thing exists?!

How do you define "cluster point"?

Homework Helper
Gold Member
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.

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)