The question is extremely general, but I do have a specific case I'd like to look at, and I'm hoping that some combination of specific pointers and general advice will help me out.
Consider the following sets:
$\textrm{St}_k(\mathbb{R}^n) := \{ (\bar{v_1}, … , \bar{v_k}) \in (\mathbb{R}^{n})^{k} \ | \ \bar{v_1}, … , \bar{v_n} \text{ are linearly independent in } \mathbb{R}^n \}$,
$\textrm{St}^0_k(\mathbb{R}^n) := \{ (\bar{v_1}, … , \bar{v_k}) \in (\mathbb{R}^{n})^{k} \ | \ \bar{v_1}, … , \bar{v_n} \text{ are orthonormal in } \mathbb{R}^n\}$,
$\textrm{Gr}_k(\mathbb{R}^n) := \{ U \subset \mathbb{R}^{n} \ | \ U \text{ is a subspace of } \mathbb{R}^n \text{ with dimension }k \}$.
We know $\textrm{St}^0_k(\mathbb{R}^n) \subset \textrm{St}_k(\mathbb{R}^n)$. We can give $\mathbb{R}^{nk}$ the standard product topology (which is easy to visualize), and thus we can give the Stiefel manifolds, $\textrm{St}^0_k(\mathbb{R}^n)$ and $\textrm{St}_k(\mathbb{R}^n)$, the subspace topology (not easy to visualize...for me anyway). Moreover, we have a surjective map from either of the two types of Stiefel manifolds into the Grassmannian, $\textrm{Gr}_k(\mathbb{R}^n)$, which sends a linearly independent set to its span; the same quotient topology arises from either of these maps (this topology is especially hard for me to visualize).
My questions are:
How do I visualize the open (or even just basic open sets) of these spaces? Are there any theoretical tools/big theorems I can use to better understand the topologies on these spaces (without necessarily having the geometric intuition for them)? Are there any patterns of thinking I should follow whenever I have trouble figuring out what open sets of spaces look like, in general?
My exposure to topology thus far has been pretty elementary, so I'm hoping the answers won't presuppose too much knowledge (although I'm ready to study as much as I need to to understand them).
Thanks in advance for the help!