Suppose $X$ is a space with a one point compactification $f: X\to Y$, where $x_0$ is the point that gets added to $X$, and suppose we have another one point compactification $f': X\to Y'$ where $x_1$ is the point that gets added to $X$.
I think we can define a homeomorphism $g: Y\to Y'$ in the following way: $g(x_0)=x_1, g(x)=f'f^{-1}(x)$ for all other $x$. Intuitively I'm thinking this should be a homeomorphism (if it's not correct me), but I'm not seeing how to prove it. I think it's fairly clear that it should be bijective, but how to show continuity (of both it and its inverse) isn't coming to me. If anyone could give a hint that would be swell.