# Is this map from a direct limit of topological spaces inductively irreducible?

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).