I am attempting to show that no point $(x,y)$ on the unit circle is a cut point.
What I have been trying to do. Let $(x,y)\in S^1$ and let $X=S^1-${$(x,y)$}$=U\cup V$ where $U,V$ are nonempty proper disjoint open subsets. Then I choose an arbitrary point in each and attempt to get a contradiction. For instance let $(a,b)\in X$ thus either $a\not=x$ or $b\not=y$. WLOG assume $a<x$.
This is where I get stuck. Do I need to set up cases? Should I be using some neighborhood technique. Any help would be very much appreciated. Thank you
From a to b there is arc ab and arc ba . x distinct from a and b is on only one of these arcs and thus is not a cut point.