I have two circles joined by a straight line, which are path-connected subsets of the plane.
If I remove any point from either circle, how many path components would there be?
Intuitively, I think there could be the same number as a circle, namely 1, but I'm not sure this is correct...?
So to show this subset is not homeomorphic to o- (where the line s the circle and the end point of the line is included) I need to show that they have different types of cut points. But, o- has infinitely many cut points of type 1(any point on the circle) and infinitely many cut points of type 2(anypoint on the line).
So via this method, how do I show they aren't homeomorphic??