Clarification on Grassman manifold. I am confused about the definition of a Grassman manifold. Suppose we have $G(n, k),$ a Grassman manifold - that is the space of $k$-dimensional subspaces of $\mathbb{R}^n$. 1) What exactly is the topology on this manifold? Apparently the topology is induced by the map $G(n, k) \rightarrow End(\mathbb{R}^n)$. I understand that means it is the coarsest topology such that the map is continuous but what is the topology on $End(\mathbb{R}^n)$? Would it be the topology induced by the sup-norm metric? 2) Now suppose we have a sequence $1 \leq i_1 < \ldots < i_k \leq n.$ Now consider $G_{i_1, \ldots, i_k}(n, k)$ a subset of $G(n, k)$ composed of subspaces whose projection onto the $i_1, \ldots, i_k$ coordinates is non-degenerate. What does non-degenerate mean here?

$\operatorname{End}(\Bbb R^n)$ is exactly the space of $n\times n$ matrices. Since this is a finite-dimensional vector space, any two topologies induced by a norm on it are equivalent, and in particular, you may think of the usual topology here.

As for the second question, non-degenerate probably means not the zero vector space (though authors can write whatever they want, and providing the context would help us to be more sure). What this means is that the given space consists of $k$-planes that have a positive-dimensional intersection with the subspace given by the indicated coordinates.