Is the number of cut points the same if they are homeomorphic. What I mean, if you had a structure that had only one 4-cut point, and then another topology with five 4-cut points. Can they ever be homeomorphic?
I know they to topologies have to have the same path components and this implies they have say a topology with 10-cut point isn't homeomorphic to one without a 10-cut point. However, I have strong intuition that its even stronger than this and the number of cut points matter. However, I can't prove it.