TITLE: A property of the unit circle
QUESTION [28 upvotes]: 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) ??

REPLY [23 votes]: This property indeed characterizes the circle, but this is not obvious.
This was shown by R. L. Moore, according to Sam Nadler's Continuum Theory p. 156.
Added: the precise reference is [522] in this historical survey of continuum theory.

REPLY [14 votes]: Yes and this is a nontrivial result, see e.g. J. G. Hocking, G. S. Young, "Topology", page 55, theorem 2-28. See also their theorem 2-27 for topological characterization of the interval.