According to this research in $\pi$-base, the sigma product of incountably many copies of $\mathbb{R}$ and the boolean product topology on $\mathbb{R}^{\omega}$ are connected, $T_2$ and homogeneous.

However, they are not first countable, hence they cannot be homeomorphic to subspaces of $\mathbb{R}^{\omega}$.

According to this search in $\pi$-base, the sigma product of incountably many copies of $\mathbb{R}$ (no longer in pi-Base as of 2023 March) and the boolean product topology on $\mathbb{R}^{\omega}$ are connected, $T_2$ and homogeneous.

However, they are not first countable, hence they cannot be homeomorphic to subspaces of $\mathbb{R}^{\omega}$.

According to this research to $\pi$-base, the sigma product of incountably many copies of $\mathbb{R}$ and the boolean product topology on $\mathbb{R}^{\omega}$ are connected, $T_2$ and homogeneous.

However, they are not first countable, hence they cannot be homeomorphic to subspaces of $\mathbb{R}^{\omega}$.

According to this research in $\pi$-base, the sigma product of incountably many copies of $\mathbb{R}$ and the boolean product topology on $\mathbb{R}^{\omega}$ are connected, $T_2$ and homogeneous.

However, they are not first countable, hence they cannot be homeomorphic to subspaces of $\mathbb{R}^{\omega}$.