Let $m$ and $n$ be positive integers and $0 \leq k \leq \min\{m,n\}$ an integer. Prove or disprove: The subspace of $M_{m \times n}(\mathbb{C})$ consisting of all matrices of rank equal to $k$ is connected. (You may use the following fact: For $t \geq 2$, $\mathrm{GL}_{t}(\mathbb{C})$ is connected.)