I want to show these two objects live up to their name in the sense that they actually are manifolds. The Grassmann manifold I understand to be a generalization of projective space (everything is done over $\mathbb R$ here), where we consider $k$-dimensional linear subspaces of $\mathbb R^n$, while the Stiefel manifold is defined as the space of all $k$-frames in this same space. By a $k$-frame I mean an ordered set of $k$ linearly independent vectors.
I have already been able to show that the Grassmann manifold actually is a manifold by using a few properties of topological manifolds to prove Hausdorf property and second-countability.
On one hand, I feel like it should be possible to go through a highly similar process to check all the conditions for the Stiefel manifold. On the other hand, it is pretty easy to see there exists a natural relationship between these manifolds - every $k$-frame spans a $k$-dimensional linear space, and so every element of a Stiefel manifold can be associated in this way with a matrix (and hence an equivalence class of these matrixes) in the Grassmann manifold.
Is there a way I can use this relationship to make short work of this question? It feels like a natural thing to do, but I don't quite see how. My study of manifolds and differential geometry is just starting, so I don't have many tools yet, just the basic properties of smooth manifolds and differentiable structures.