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.
Are 2 diagram homeomorphic?
2
$\begingroup$
general-topology
1 Answers
4
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).
-
0I 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? – 2012-12-14
-
0@Clayton: Yes, that would also work: $4$ non-cut-points in the one, $6$ in the other (assuming that the line segments are closed). – 2012-12-14
-
0@BrianM.Scott so NUMBER of component sets is also a topological properties right? – 2012-12-14
-
0@Mathematics: Yes, it is: homeomorphisms take connected sets to connected sets. – 2012-12-14