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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

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

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    Hint: Do you know that the real line is connected? – 2017-02-21
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    Yes but, I am actually planning to use this to show that the unit circle is not homeomorphic to the real line. – 2017-02-21
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    Hint to the hint: My hint is in connection to $S^1 - \{(x,y)\}$, not to $S^1$ itself. – 2017-02-21
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    So i should try to show that this set is homeopmorphic to the real line? – 2017-02-21
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    Yes. ${}{}{}{}$ – 2017-02-21

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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.