TITLE: Which topological spaces contain dense simply connected subspace?
QUESTION [7 upvotes]: 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.

REPLY [3 votes]: Any real semialgebraic set $X \subset \mathbb{R}^N$ has a dense, open subset that is a submanifold: just take the complement of its singular set. In fact, the singular set is Zariski closed in $X$, hence nowhere dense in the euclidean topology of $X$. 
Using your remark about manifolds, it follows that any semialgebraic set contains an open, dense subspace homeomorphic to $\mathbb{R}^n$, with $n = \dim X$.