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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.

enter image description here

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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).

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I believe one could also say there are 4 points which don't disconnect the set on the left, while there are 6 such points on the right diagram. Is that correct? –  Clayton Dec 14 '12 at 5:40
    
@Clayton: Yes, that would also work: $4$ non-cut-points in the one, $6$ in the other (assuming that the line segments are closed). –  Brian M. Scott Dec 14 '12 at 5:41
    
@BrianM.Scott so NUMBER of component sets is also a topological properties right? –  Mathematics Dec 14 '12 at 5:57
    
@Mathematics: Yes, it is: homeomorphisms take connected sets to connected sets. –  Brian M. Scott Dec 14 '12 at 6:08

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