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What is 'cluster point' in french?

  1. May 28, 2007 #1

    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).
     
  2. jcsd
  3. May 28, 2007 #2
    I believe it's "aaleur d'adhérence". All I did is wikipedia "Cluster Point" and then I clicked "français" on the left.
     
  4. May 28, 2007 #3

    quasar987

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    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!
     
  5. Jun 1, 2007 #4
    Point d'accumulation
     
  6. Jun 1, 2007 #5

    quasar987

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    '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, [itex](B_{\epsilon}(a)\backslash \{a\})\cap A\neq \emptyset[/itex] (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: Jun 1, 2007
  7. Jun 1, 2007 #6
    Have you considered peeking at a math french/english dictionary at some library?
     
  8. Jun 1, 2007 #7

    quasar987

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    This thing exists?!
     
  9. Jun 1, 2007 #8
    How do you define "cluster point"?
     
  10. Jun 1, 2007 #9

    quasar987

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    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.
     
  11. Jun 2, 2007 #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)
     
  12. Jun 2, 2007 #11

    quasar987

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    I don't see how this helps but nice site, thx.
     
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