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All spaces are separable metrizable.

Let $f:X\to Y$ be a continuous open mapping from a Polish space $X$ onto a zero-dimensional space $Y$. By the Michael selection theorem (zero-dimensional version), the multi-valued pre-image mapping $f^{-1}:Y\to X$ has a continuous selection. In particular, this shows that $Y$ contains a topological copy of $X$.

My question is whether a similar theorem can be proved when $X$ is a complete almost zero-dimensional space such as $\mathfrak E_c:=\{\mathbf x\in \ell^2:x_n\in \mathbb R\setminus \mathbb Q\text{ for all }n\}$ or its $\omega$-power. Note: These spaces are homeomorphic to graphs of lower semi-continuous functions with zero-dimensional domains.

It would imply that the homeomorphism group of the pseudo-arc has positive dimension: Let $P$ be the pseudo-arc. Let $Y\subset P$ be a copy of $\mathfrak E_{c}$ spaces above. Let $\mathcal H(P)$ denote the homeomorphism group of $P$. Fix $x\in P$ and define $g:\mathcal H(P)\to P$ by $f(h)=h(x)$. Then $g$ is continuous and open (by Effros) and maps onto $P$ (since $P$ is homogeneous). Let $X=g^{-1}[Y]$ and let $f=g\restriction X$. Then $f$ is continuous and open, and $X$ is complete so the selection theorem would apply, and thus $\mathcal H(P)$ contains $\mathfrak E_c$, a space of positive dimension.

The dimension of $\mathcal H(P)$ is still one of the biggest open questions in continuum theory. This is just one approach that I thought might work.

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