Let $A,B$ be matrices in $GL(n,\mathbb{R})$ sufficiently close in the usual metric on matrices. Suppose $A$, resp., $B$ stabilizes a $k$-dimensional subspace $U$, resp., $V$ of $\mathbb{R}^n$, where $0 < k < $ dim $V$. Is there any topological relationship between $U$ and $V$? Is there any metric on the set of $k$-dimensional spaces, and if Yes can it be said that $U$ and $V$ are close in that metric?