# A property of the unit circle

Let $(X,d)$ be a compact connected metric space with the property that for any distinct points $a,b$, $X\backslash \lbrace a,b\rbrace$ is disconnected. Clearly the unit circle has this property. Is there any other example (up to isomorphism) ??

• Is a circle with a twist (so $\infty$) a "compact connected metric space"? – Mark Hurd Aug 3 '13 at 17:22
• Yes, but it is not disconnected by any two points. And, for a more difficult attempted counterexample, the one point compactification of the closure of the graph of $\sin(1/x)$ (a cousin of the so called Warsaw circle), isn't either. – BS. Aug 3 '13 at 20:40

• @BS: Thanks for the interesting reference. The question is also related to the old Phragmen-Brouwer property (PHB) of a space $X$: This is that $X$ is connected and and if $D,E$ are disjoint non-empty closed subsets of $X$ and $a,b$ are points of $X\backslash (D\cup E)$ which lie in the same component of $X \backslash D$ and in the same component of $X \backslash E$ then $a,b$ lie in the same component of $X \backslash (D \cup E)$. If $X$ does not have the PHB, and is path connected and locally path connected, then its fundamental group contains the integers as a retract! – Ronnie Brown Aug 4 '13 at 16:08