If x is an accumulation point of set S and e >0

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Discussion Overview

The discussion revolves around the concepts of accumulation points and limit points in the context of set theory. Participants explore the relationship between these two definitions, particularly in relation to neighborhoods around a point and the presence of other points within those neighborhoods.

Discussion Character

  • Conceptual clarification, Debate/contested

Main Points Raised

  • One participant proposes that proving "if x is an accumulation point of set S and e >0, then there are infinite number of points within e of x" is equivalent to stating "if x is a limit point of set S, then every neighborhood of x contains infinitely many points of S."
  • Another participant agrees with this equivalence.
  • A different participant suggests that a "limit point" must be an accumulation point for the domain, indicating a potential distinction between the two terms.
  • One participant defines an accumulation point as a point in set A where any neighborhood of x contains at least one other point in A that is not x, although this definition is repeated and not universally accepted.
  • Another participant reiterates the definition of an accumulation point, suggesting that it may not be necessary to differentiate between the terms in this context.

Areas of Agreement / Disagreement

There is some agreement on the relationship between accumulation points and limit points, but the discussion reveals uncertainty and potential disagreement regarding the precise definitions and implications of these concepts.

Contextual Notes

Participants express varying levels of familiarity with the definitions, and there is a lack of consensus on the necessity of distinguishing between accumulation points and limit points.

irony of truth
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If I were proving that "if x is an accumulation point of set S and e >0, then there are infinite number of points within e of x", is it exactly the same as saying "if x is a limit point of set S, then every neighborhood of x contains infinitely many points of S"?
 
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Seems so to me.
 
Well, a "limit point" (I cannot find the precise meaning of the phrase) has to be an accumulation point for the domain, then...
 
As I recall an accumulation point is a point in the set A where any neighborhood of x has at least one other point in A that is not x.
 
MalleusScientiarum said:
As I recall an accumulation point is a point in the set A where any neighborhood of x has at least one other point in A that is not x.

Not necessary.
 

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