if A is an open subset of R, then A is dense if and only if R - A is nowhere dense

i am trying to prove the => direction first.

assuming that A is dense, i know that

i also know that A is open so

the trouble i am having is that i am not sure if because if that was true then it follows that is nowhere dense.

could someone help me continue the proof? thanks.

Re: if A is an open subset of R, then A is dense if and only if R - A is nowhere dens

Since is open, is open and you can choose such that .

Re: if A is an open subset of R, then A is dense if and only if R - A is nowhere dens

Quote:

Originally Posted by

**oblixps** i am trying to prove the => direction first.

assuming that A is dense, i know that

i also know that A is open so

the trouble i am having is that i am not sure if

because if that was true then it follows that

is nowhere dense.

is an open set. Is it not?