I'm trying to figure out a problem here and I don't quite get it.

Let

be a bijection between

and

. For

, define

to be the open set

-ball around the point

. Define

. Show that F is compact.

Now I don't want the answer to this question; I want to figure it out. It's just that I could swear that the set

is empty, which I know is compact. However, the next part of the question asks me to show that

and that

.

Basically, it looks to me like

completely engulfs

and then some, so

wouldn't contain any values. Thus the set F is empty.

Please Help! Thank you very much.