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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