First of all we should notice two facts:

1.

2.

Ad 1.

It follows from

Ad 2.

It follows from

Now, since is open, is closed.

Since closure of A is the smallest closed set containing A from 1 we have:

Now, since is closed, is open.

Since interior of A is the largest open set in A we from 2 we have

Thus:

So we have: