The following assertion appears in a paper I am reading, and I can't seem to verify it.

Let $\text{Gr}_{n,m}$ denote the set of pairs $(V,W)$ where $V$ and $W$ are as follows.

- $V$ is an $n$-dimensional subspace of $\mathbb{C}^{\infty}$.
- $W$ is an $m$-dimensional subspace of $\mathbb{C}^{\infty}$.
- $V$ and $W$ are orthogonal.

The space $\text{Gr}_{n,m}$ has an obvious topology. If $\text{Gr}_n$ and $\text{Gr}_m$ are the usual Grassmannians of $n$ and $m$ planes in $\mathbb{C}^{\infty}$, then there is an obvious map $\psi : \text{Gr}_{n,m} \rightarrow \text{Gr}_n \times \text{Gr}_m$.

The map $\psi$ is almost a homeomorphism, but not quite because of condition 3 above. The paper claims that $\psi$ is a homotopy equivalence.

Thanks for any help!