How do i prove that these 2 path connected subsets are not homeomorphic?
I have the following 2 path connected subets:
(Imagine that the line intervals are connected to the circles in each case, i just wasnt able to draw them touching but they are)
and for part (i) the end point of the lin interval IS included in the subset.
How can i prove that subset (i) is NOT homeomorphic to subset (ii) ?
I thought first that this involves studying the cut points and showing that they have different numbers of cut points of each type.
. (i) and (ii) both have infinite cut points of type 1
. they both have infinite cut points of type 2
. they both have zero cut points of type 3,4,...
So they have the same number of cut points of each type which doesnt help me.
Then i thought i must have to study cut pairs.
. (i) and (ii) have infinite cut pairs of type 1
. they both have infinite cut pairs of type 2
. they both have infinite cut pairs of type 3
. then i don't think its possible to find any cut pairs of type 4 in either (i) or (ii).
So they have same number of cut pairs of each type...no use.
Can anyone help show me how they are not homeomorphic?
Re: How do i prove that these 2 path connected subsets are not homeomorphic?
Would anyone like to elaborate on this?
I understand that both i and ii have infinitely many 1-point and 2-point cutpoints. Would you have to look at cut-pairs in this example? Or is there another way of going about it?