$\sigma$-product of the Hilbert cube

Question

Given a homogeneous space $X$ and $p\in X$, we define the sigma product to be the following subspace of $X^\omega$: $$\sigma X=\{\mathbf x \in X^\omega:x_n=p\text{ eventually}\}$$

("eventually" means "for all but finitely many $n$", or "for all $n$ sufficiently large).

I am interested in cases when $\sigma X$ is homeomorphic to $\mathbb Q\times X$. In order for $X$ to have this property, its dimension must be either $\leq 1$, or infinite. Some $0$-dimensional examples are $X=\mathbb Q, \mathbb P, \mathbb Q ^\omega$. Some $1$-dimensional examples are the Erdos spaces $X=\mathfrak E, \mathfrak E_{\mathrm{c}}$, which are the set of all points in the Hilbert space $\ell^2$ whose coordinates are all rational, or irrational, respectively.

My question is about the existence of infinite dimensional examples.

Question. Is $\sigma X$ homeomorphic to $\mathbb Q\times X$ for $X=[0,1]^\omega$ or $X=\mathbb R ^\omega$?

Answer 1

Let $X = \sigma({\mathbb{Q}} \times [0,1])$, where $p$ is the point which is $(0,0)$ in each coordinate. Then $\sigma X$ is homeomorphic to $X$ since each consists of all points which are unequal to $p$ only finitely often. Therefore, it is enough to show that $X$ is homeomorphic to ${\mathbb{Q}} \times X$. Since ${\mathbb{Q}} \times {\mathbb{Q}}$ is homeomorphic to ${\mathbb{Q}}$, the space obtained from $X$ by replacing the first factor in the product with $({\mathbb{Q}} \times {\mathbb{Q}}) \times [0,1]$ gives a space homeomorphic to $X$, that is, ${\mathbb{Q}} \times X$ is homeomorphic to $X$, which is homeomorphic to $\sigma X$. (If obvious changes are made to $p$, instead of $[0,1]$, you can use $[0,1]^\omega$ or ${\mathbb{R}}$or ${\mathbb{R}}^\omega$ or other reasonable spaces.)