Why is this subset closed I try to understand the following statement:
Let $L(\mathbb{R}^n,\mathbb{R}^m)$ denote the vector space of linear maps from $\mathbb{R}^n$ to $\mathbb{R}^m$. Now the statement is that the set
$S := \{\phi \in L(\mathbb{R}^n,\mathbb{R}^{m-k})~|~ \text{rank}(\phi)<m-k \}$ is a closed subset in $L(\mathbb{R}^n,\mathbb{R}^{m-k})$.
Can anyone tell me why that is? Thanks!

Its complement is the set of all invertible linear maps. There's a general fact:

For any Banach spaces $E$ and $F$, the set of invertible linear maps $E \to F$ is open in $\mathcal L(E,F)$ for the topology of $\|\cdot\|_{op}$.

Where $\|\cdot\|_{op}$ is the operator norm.
To prove this, you need Von Neumann's lemma. It's explained here how that is done.
