# On Grassmann manifold

In most references about Grassmann manifold, we usually introduce the following map: suppose that $$S$$ is a fixed subspace of codimension $$k$$, and let $$\Omega_S \subset Gr(k,\mathbb{K})$$ be all subspaces $$W$$ such that $$W \cap S = \{0\}$$. $$\Omega_S$$ is an open set.

Fix some $$U \in \Omega_S$$. One can show that $$W \in \Omega_S$$ if and only if there exists some linear map $$\beta_W \in Hom(U,S)$$ such that $$W = \{x + \beta_W x : x \in U \}$$. From there, one can easily check that there exists a bijection $$\Omega_S \simeq Hom(U,S)$$.

I'm interested in show that this bijection is actually an homeomorphism of topological spaces!

So far, I haven't been able to make a lot about this claim. What bugs me is that the topologies at stake are not necessarily intuitive to work with: $$Gr(r, \mathbb{K})$$ is endowed with the quotient topology, and $$Hom(U,S)$$ is really just an euclidean space, so might just take some random norm/distance.

Any help greatly appreciated !