@hugh Thomas: I agree that you can use this approach to show that the answer to my question is "an open subset of the Grassmannian of n+1-dimensional linear subspaces of R^{m+1}", but I don't believe this uniquely identifies a manifold.

@hugh Thomas This is close, but the grasmannian is related to linear subspaces, not the set of all flats of a certain dimension. So, any flat that doesn't pass through the origin would be excluded.

@Thierry Zell: Yes. Sorry. @Andres Caicedo: I suppose that since we don't have a definition of 'limit' for lines, 'continuous' would not be rigorously defined . Here is the definition I am using: a sequence of lines L(1), L(2), etc. is said to approach another line L if, for any point p on L, the limit as n goes to infinity of the minimum distance between p and L(n) is 0.

@Allen Knutson: As Steven and Simon pointed out, (a,b) |-> {y = ax+b} is not one to one. It misses the case y=c. @Steven: You are completely correct; my apologies. By 1:1, I meant that the function was bijective, but apparently the standard defenition of 1:1 is injective, so that is my fault. I edited the problem so that it was correct.