# Nowhere dense subset of a metric space

1. Feb 18, 2010

### de_brook

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

2. Feb 18, 2010

### wofsy

just take a Cauchy sequence without its limit point e.g. 1/2^n

3. Feb 25, 2010

### de_brook

Is zero not a limit point of 1/2^n since as n gets large, 1/2^n goes to zero?

4. Feb 25, 2010

### WWGD

"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.