Suppose that $X$ is a metric continuum irreducible between two points $p$ and $q$.
Suppose further that whenever $U$ is a connected open set missing $p$ and $q$, we have $X\setminus U$ has two connected components, one containing $p$ and the other containing $q$.
Is $X$ necessarily equal to (homeomorphic to) $[0,1]$?
EDIT: I don't have much context to add. I came up with this question while trying to solve a very different problem. Honestly I have to clue if the answer is yes or no, though if the answer is yes it would be useful to me.