Nowhere dense subset of a metric space

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SUMMARY

A nowhere dense subset of a metric space can indeed be non-closed, as illustrated by the example of the sequence 1/2^n, which converges to zero. In this case, zero is not a limit point of the set {1/2^n} because it is not included in the set itself. The discussion emphasizes that a closed subset must contain all its limit points, reinforcing the definition of closed sets in metric spaces and topological spaces.

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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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