And when can this subspace be chosen to be open?

As the answer to this question indicates, any manifold contains an open dense subset, which is homeomorphic to $\mathbb{R}^{n}$, and so for manifolds the stronger of the two properties holds. I wonder how widespread this phenomenon is.

Note that simply connected spaces are assumed to be connected, and so we only consider connected topological spaces. Also feel free to narrow the class of spaces further, e.g. add local path connectedness, local compactness, etc.