The balls B(x,r)={y : |x-y|<r} are open sets.
Maybe this helps already?
Suppose has the following property: For any open set , is an open set in . Show that is continuous.
I am completely stuck on this. I know that there is a connection between the definition of continuity and how if is an open set, then given , there exists such that but I am missing the connection and can't really formalize any sort of proof. Thanks in advance.
Hmm, thanks, I think that does help. So here is my attempt.
Given and , let , which is clearly an open set . Since , we have that . So is an interior point of , which means there exists such that implies , which in turn implies . Since implies , we have that given , there exists such that implies , which means is continuous at . Since was arbitrary, we have that is continuous on . Q.E.D.