According to [this research][1] to $\pi$-base, the [*sigma product of incountably many copies of* $\mathbb{R}$][2] and the [*boolean product topology on* $\mathbb{R}^{\omega}$][3] are connected, $T_2$ and homogeneous. 

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


  [1]: https://topology.jdabbs.com/spaces?q=(~%20First%20Countable%20%2B%20%24T_2%24%2B%20connected%20%2B%20homogeneous)
  [2]: https://topology.jdabbs.com/spaces/S000157
  [3]: https://topology.jdabbs.com/spaces/S000107