# Selection theorems and homeomorphism groups

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.