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Accumulation/densification points.

  1. Feb 15, 2005 #1

    Galileo

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    I noticed a small difference in two definitions I thought were equal.
    We have two different words for these points in Dutch. Can someone tell me the equivalent english terms?
    These definitions are strictly in the field of Topology (no metrics or sequences or limits allowed).

    Let X be a topological space and A a subset of X.

    Accumulation point (translated from Dutch 'ophopingspunt'):
    A point [itex]x \in X[/itex] is called an accumulation point of [itex]A[/itex] if every open neighbourhood of [itex]x[/itex] has a non-empty intersection with [itex]A[/itex].

    This one's a toughy to translate. Here are a few attempts:
    Densification point - Aggregation point (translated from Dutch 'verdichtingspunt'):
    A point [itex]x \in X[/itex] is called a 'densification' point of [itex]A[/itex] if every open neighbourhood of [itex]x[/itex] contains a point of [itex]A[/itex] unequal to [itex]x[/itex].

    BTW: I think I've seen these being mixed up in Dutch as well...

    Anyone?
     
  2. jcsd
  3. Feb 15, 2005 #2

    mathman

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    There is a difference when x is in A, but is an isolated point. In this case x is an accumulation point, but not a "densification" point.
     
  4. Feb 15, 2005 #3

    Galileo

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    I know. I asked if someone knew the equivalent english names. :wink:
     
  5. Feb 15, 2005 #4

    mathwonk

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    well topologists use mostly the concept of accumulation point, and do not use the other concept at all nowadays, at least in my opinion, (not a topologist though).

    analysts however, being old fashioned, still use the second concept and call it "limit point".

    this causes topology students some difficulty when taking prelim exams written by analysts.
     
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