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Let $q:Y\to X$ be a continuous surjective map between topological spaces. Then $q$ is inductively irreducible (Gruenhage) if $Y$ has a closed subset $V$ such that $q|V$ maps $V$ surjectively onto $X$ and no proper subset of $V$ has this property (in other words, $q|V$ is irreducible).

Let $Y_1=[0,1)$ and $Y_{n+1}=Y_n\times Y_1$ $(n\ge 1)$. Let $Y$ be the disjoint union of the spaces $Y_n$. Define an equivalence relation on $Y$ by identifying $y\in Y_n$ with $(y,0)\in Y_{n+1}$ at each stage. Let $q:Y\to X$ be the quotient map. Then I think that $X$ is the direct limit of the spaces $Y_n$ (not that it matters for this question).

Question: Is $q:Y\to X$ inductively irreducible? (I am hoping that the answer is no).

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