Nowhere dense subset of a metric space

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Discussion Overview

The discussion revolves around the properties of nowhere dense subsets in metric spaces, specifically focusing on whether such subsets can be non-closed. Participants seek examples and clarify the implications of limit points in this context.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants inquire about examples of nowhere dense subsets that are not closed.
  • One participant suggests using a Cauchy sequence, specifically the sequence 1/2^n, as an example of a nowhere dense subset that is not closed.
  • Another participant questions whether zero is a limit point of the sequence 1/2^n, noting that the sequence approaches zero as n increases.
  • A subsequent reply confirms that zero is indeed a limit point of the sequence and discusses the implications of limit points in relation to closed sets in metric spaces.
  • There is a mention of a general property of closed subsets in topological spaces regarding limit points, but this is presented without consensus on the specific example discussed.

Areas of Agreement / Disagreement

Participants express differing views on the nature of limit points and their relation to closed sets, indicating that the discussion remains unresolved regarding the example of the Cauchy sequence and its properties.

Contextual Notes

The discussion does not resolve the question of whether the sequence 1/2^n serves as a valid example of a nowhere dense subset that is not closed, as the implications of limit points are still being debated.

de_brook
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Can we have some examples in which a nowhere dense subset of a metric space is not closed?
 
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de_brook said:
Can we have some examples in which a nowhere dense subset of a metric space is not closed?

just take a Cauchy sequence without its limit point e.g. 1/2^n
 
Is zero not a limit point of 1/2^n since as n gets large, 1/2^n goes to zero?
 
"Is zero not a limit point of 1/2^n since as n gets large, 1/2^n goes to zero? "

Yes, and that is precisely the issue here. A closed subset of a metric space

(I think this is true in any topological space)contains all its limit points. One

way of seeing this is seeing what would happen if the limit point L of a closed

set C in X was not contained in C. Then L is in X-C, and every 'hood (neighborhood)

of L in X-C , intersects points of C.
 

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