How can I visualize what open sets "look" like in unfamiliar topological spaces? 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!
 A: There are several intuitive approaches to topology, namely:


*

*Intuition coming from open sets: includes neighbourhoods.

*Intuition coming from closed sets: includes limit points.

*Intuition coming from continuous maps of a model topological space to the space in question: includes paths and embeddings.

*Intuition coming from continuous maps of the space in question to simpler spaces: includes continuous functions (real-valued or else) and projections.


Note that neighbourhoods’ approach never requires to imagine any possible neighbourhood – we can limit ourselves to a convenient base of the topology (open balls, open intervals and their products, etc.).
Neither of these approaches requires a significant mental strain for the case of Stk: all four concepts can be implemented as k neighbourhoods around each of vectors not intersected simultaneously by any linear k − 1-space, componentwise limits, k paths with necessary restrictions, and a function of k points respectively.
Grk is not very harder: linear subspaces avoiding some compact set enclosing U, convergence of subspaces (that are closed subsets and even compacts within closed unit ball), a system of linear equations (specifying U) whose coefficients depend on an additional parameter, and a continuous functional on linear operators from U to U⊥ (i.e. local coordinate system on the Grassmannian) respectively.
You will encounter a partial problem with St0k (a neighbourhood in orthonormal bases, what is it? ), but it can be compensated with alternative modes of visualization. For instance, St0k is a closed subset in Stk.
