Let $q: [0,1]\to X$ be an irreducible quotient map (irreducible meaning that no proper closed subset of $[0,1]$ maps subjectively onto $X$) with $X$ Hausdorff. By Whyburn's theorem (1939) the set of singleton fibres is dense in $[0,1]$. Does the set of singleton fibres contain an open subset of $[0,1]$?
(The answer is no if $[0,1]$ is replaced by the Cantor space, as someone has kindly pointed out recently: Is Whyburn's theorem on irreducible maps optimal?)