I'm looking for two non-homeomorphic connected, Hausdorff, locally compact spaces whose one-point compactifications are homeomorphic.
Without the connectedness property this is easy, for example: $[0,1) \cup (1,2]$ and $[0,2)$. I was thinking that I could maybe find two connected spaces which are locally compact but not compact where on loses the connectedness property when removing a point, and the other doesn't, but I can't seem to find such a space. Can anyone give an example of two such spaces?