Suppose is a topological space, and suppose that is an open subspace of . If is a dense subset of , prove that is dense in .

Here's my attempt at a proof (with help from other people)... I'm not certain if the argument is sound or not, or even if there is a better way to prove it. We decided to try subset inclusion...

Proof:

Suppose that the opposite is true, and let . The set is open, hence if it is nonempty it must meet the dense set . This is a contradiction, since and so , but is disjoint from . Thus, .

(I'm not completely sold on this argument... But I can't think of one, and they assure me it's right... I just don't know why).

We know that if a set is closed in , then it must equal for some . Threfore, .

Any help is greatly appreciated.