Given two spheres touching at one point, would that be a surface? I think not, but I don't know how to begin to search for the proof. I thought maybe proving it isn't Hausdorff, but I'm not clear on how to go about it. Anyone can help?

Sorry about the bad English, not my mother tongue...

I don't think so but I can't think of a proof either. Intuitively I'd say that's because you can draw Cartesian axes around that point - you could have an axis pointing at the two spheres but the the axis orthogonal to that would have nowhere to go.

Let me qualify this first by saying that I am not a mathematician but it seems to me that a point is the smallest part of a surface and not a surface in and of itself.