Indeed there are many way to prove whether something are homeomorphic with each other. For the diagram below, it seems that they are not homeomorphic but i am not sure how to argue that.

Indeed there are many way to prove whether something are homeomorphic with each other. For the diagram below, it seems that they are not homeomorphic but i am not sure how to argue that.

Removing the central point of the second diagram leaves a set with $6$ connected components; there is no point in the first diagram that has that property, and it is a topological property (i.e., one preserved by homeomorphisms).