In the proof of Bott periodicity for the unitary group in Milnor's Morse theory (Lemma 23.1, page 128), it is asserted that the space of minimal geodesics from $I$ to $-I$ in the special unitary group $SU(2m)$ is homeomorphic to the complex Grassmannian of all $m$-dimensional vector subspaces of $\mathbb{C}^{2m}$.
Prior to this, Milnor explained how every minimal geodesic comes from an initial tangent vector $A \in \mathfrak{su}(2m)$, which, as a linear endomorphism splits $\mathbb{C}^{2m}$ as the orthogonal direct sum of two eigenspaces corresponding to the eigenvalues $+i\pi$ and $-i\pi$. Each eigenspace is a $m$-dimensional subspace, so we can identify the minimal geodesic with the $+i\pi$-eigenspace, and thus obtain a correspondence between the set of minimal geodesics and the complex Grassmannian.
However, it is not obvious to me that this bijection is a homeomorphism. So I would appreciate any arguments or ideas about why this is so.
Firstly, we are equipping the space of minimal geodesics with compact-open topology, which in this case is metrizable with metric $d(\gamma,\gamma') = \max_{0 \leq t \leq 1} \rho(\gamma(t), \gamma'(t))$, where $\rho$ is the distance function induced by the Riemannian metric on $SU(2m)$, which in turn is given by $\langle A, B \rangle = \operatorname{Re} \operatorname{trace}(AB^*)$.
On the other hand, the Grassmannian has the topological structure inherited as an orbit space, e.g., as $U(2m)/U(m) \times U(m)$.
I don't have very much intuition with these topologies, but perhaps it is plausible that if two $m$-dimensional subspaces are "close", then the matrices in $\mathfrak{su}(2m)$ to which they correspond are also close, and so the geodesics are close, and vice-versa. How I can make this rigorous?
(Later, Milnor gives a similar lemma (Lemma 24.1) for the space of minimal geodesics on the orthogonal group $O(n)$ and the space of complex structures on $\mathbb{R}^n$, but as far as I can tell he gives no argument on their topological equivalence either.)