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?
I stated that the end point of the line interval for (i) IS included in the subset.
Therefore surely it is possible to remove 2 points in (i), namely the end point of the line interval and some point on the circle that it is connected to and it will still remain path connected.
Is that not true?
Indeed. I didn't see that end-point included thing. Let us try the following:
suppose f: (i) --> (ii) is a homeom. between the two given sets, and let I denote the line in (i) with both end-points included (the right one and the left one where it makes contact with the circle) ==> As f is a closed map it must be f(I) is a closed set and, in fact, homeom. to a closed bounded interval (we may think of [0,1]).
The question is: what is f(I) in (ii)? It can be one of the following options:
a) A closed arc in some of the two circles
b) The closed line joining both circles (included both end-points)
c) Part of a closed arc in one of the circles, or in both, and part (or all) the line segment joining both circles
Now, as (i) - I has two CONTRACTIBLE (path-)components, it must be that (ii) - f(I) also has two contractible components, but :
(a) cannot be since we're left with a loop
(b) cannot be since we're left with three components
(c) cannot be because we're left with
A fourth option would be one whole circle and part of the line segment (and or part of a closed arc of the second circle), but this can't be since I is contractible and something containing a whole circle isn't.
Check the above, I think it's a fair chance to work this time.
Does shape ii) not only have a finite number of cut points of type 1?
namely the points above and below where the line touches each circle.
edit: actually using your arguement about cutting anywhere along the line of i) leaves 2 path components, I can see why it is difficult to find the difference.
edit: actually you could keep taking end points for shape ii) ... posted too early.
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?