Let $W\subset \mathbf R^{k}$ be an open set. Are there conditions on $W$ that guarantee the existence of a map $T:(0,1)^k \rightarrow W$ such that: (i) $T$ is surjective, (ii) $T$ is continuously differentiable, and (iii) $T$ has bounded derivative with everywhere nonsingular Jacobian whose determinant is bounded away from zero?

My (naive?) intuition is that $W$ must be bounded, connected and have "smooth" boundary.

Alternatively - is there a characterization of the sets in $\mathbf R^k$ that are diffeomorphic to $\mathbf R^k$?

Thank you in advance!