# Diffeomorphism between a triangle and a square?

Is there always a diffeomorphism between $(0,1)^2$ and any given (not degenerate) triangle?

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Diffeo-morphism?? I would doubt. At maximum between the open inner part of the square and the triangle. – Berci Nov 28 '12 at 16:42
I don't understand your second sentence. – dcs24 Nov 28 '12 at 16:43
@Berci is referring to the fact that every convex open subsets of the Euclidean plane is diffeomorphic to the unit open ball. – Willie Wong Nov 28 '12 at 16:51
@Berci Sorry, the second sentence didn't parse. Now it makes sense. – dcs24 Nov 28 '12 at 16:55
@WillieWong Thanks. So if I take $(0,1)^2$ then I'm good? (Set of measure zero is nothing to me) – dcs24 Nov 28 '12 at 16:56

If by "triangle" you mean the open set bounded by three line segments (the boundaries themselves are not included), then yes, every convex open subset of the Euclidean plane is diffeomorphic to $\mathbb{R}^2$.
In fact every simply connected open subset of the Euclidean plane is diffeomorphic to $\mathbb R^2$. – JSchlather Nov 28 '12 at 17:01