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 $\mathbb Q\times X$. It is clear that 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 $X=\mathbb R ^\omega$?