Definition. A closed subset $S$ of a topological space $X$ is called a separator between points $x,y\in X\setminus S$ if the points $x$ and $y$ belong to different connected components of $X\setminus S$. A separator $S$ is called an irreducible separator between $x$ and $y$ is $S$ coincides with each closed separator between $x$ and $y$ that is contained in $S$.
Using Kuratowski-Zorn Lemma it is easy to prove that each closed separator in a locally path-connected (or more generally locally continuum-connected) space contains a closed irreducible separator. Is the case result true without the local path connectedness?
Question. Does each closed separator between two points $a,b$ of a metrizable continuum contain an irreducible closed separator between $a$ and $b$?
I hope that the answer to this question should be know but somehow I cannot find in the books of Kuratowski and Nadler.