Simple question relating diffeomorphisms and homeomorphisms.

[Pinu7]

Consider a Euclidean space or a manifold or whatever. Furthermore, consider two regions on this space. If one can construct a diffeomorphism between the points from one region to the other, does this imply that the two regions are homeomorphic?

My gut feeling is "yes," but I would like a confirmation with maybe an explanation.

 

[Tinyboss]

Every diffeomorphism is in particular a homeomorphism, yes.

 

[homology]

though clearly not the converse. To have a diffeomorphism you need some sort of differentiable structure which an arbitrary topological space does not have.

 

[mathwonk]

if i have ham and eggs, does that mean i have eggs? i.e. you could only ask this question if you do not know what the words in it mean.

 

[quasar987]

The explanation is the basic fact that a differentiable function is continuous.

 

[Landau]

if i have ham and eggs, does that mean i have eggs? i.e. you could only ask this question if you do not know what the words in it mean.

Ha, nice answer :)