I have a few questions - they all look very straightforward but I haven't actually got answers to them and it's greatly annoying me...

Let (X,T) be a Hausdorff topological space and let A be a non-empty subset of X.

1) Prove that if an open set U in T has non-empty intersection with $\displaystyle \overline A$, where $\displaystyle \overline A$ is the closure of A, then U has a non-empty intersection with A.

2) Prove that A' is closed in A, where A' is the set of limit points of A. (Note, if we can prove that A ' contains all its limit points then we're done, but just because A ' contains all of the limit points of A it doesn't necessarily mean it contains all the limit points of A'...)

3) Prove $\displaystyle (\overline A)' = A'$