I want to prove that The complement of any set of first category on the line is dense iff the intersection of any sequence of dense open sets is dense. but I don't need to prove this in general. I need to prove that in R and I want to prove this using the following definitions: A set A is dense in the interval I if A has a nonempty intersection with every subinterval of I. It is called dense if it is dense in the line R. A set A is nowhere dense if it is dense in no interval, that is, if every interval has a subinterval contained in the complement of A.
I have proved that if the complement of any set of first category on the line is dense then the intersection of any sequence of dense open sets is dense.
on the other side: I tried to prove that like this:
suppose that the intersection of any sequence of dense open sets is dense. let A is a set of first category. we have to prove that $A^c $ is dense. since A is a set of first category then where $ \{A_n\}_{n\in N} $ is a sequence of no where dense sets $ A= \bigcup_{n \in N} A_n$ then $ A^c = \bigcap_{n \in N} A_n^c $
since $ A_n $ is no where dense then $ A_n^c $ is dense.
if I prove that $ A_n^c $ is open then the prove is done. but I can't see that.