Homemorphism of two metric space

[xiavatar]

If I create a bijective map between the open balls of two metric spaces, does that automatically imply that this map is a homemorphism?

 

[Jorriss]

Yeah, it does (if I'm interpreting your statement correctly). You only need to check continuity on a basis, so if you have a bijection $f: X \to Y$ between two metric spaces and it is continuous on a basis for $Y$ and an open map on a basis for $X$ it's a homeomorphism.