How do i prove that these 2 path connected subsets are not homeomorphic?

I have the following 2 path connected subets:

(i) O-

(ii) O-O

(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.

However:

. (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.

But:

. (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?

Thanks

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?