I think I see my error...

I was thinking of it as a subspace of

Our Space

\

Since this is the whole space its compliment must be empty

also

before I claimed that

I think this is incorrect

X is open because for every point

There exits and

such that a Ball centered at x of radius epsilon

The Emptyset is also open

Since they are compliments of each other they are also closed.

I'm sorry if I haven't been very helpful. I guess I don't remember this as well as I thought. Time for more review.

I hope this helps.

Good luck.