  For any $n\geq k\geq l\in\mathbb{N},$ let $S(n,k,l)$ be the set of all those
non-negative definite matrices $a\in M_{n}(\mathbb{C})$ with $l\leq\text{rank
}a\leq k$. Motivated by applications to $C^{*}$-algebra theory, we investigate
the homotopy properties of continuous maps from a compact Hausdorff space $X$
into sets of the form $S(n,k,l).$ It is known that for any $n,$ if $k-l$ is
approximately 4 times the covering dimension of $X$ then there is only one
homotopy class of maps from $X$ into $S(n,k,l)$, i.e. $C(X,S(n,k,l))$ is path
connected. In our main Theorem we improve this bound by a factor of 8. By
combining classical homotopy theory methods with $C^{*}$-algebraic techniques
we also show that if $\pi_{r}(S(n,k,l))$ vanishes for all $r\leq d$ then
$C(X,S(n,k,l))$ is path connected for any compact Hausdorff $X$ with covering
dimension not greater than $d$.
