  For any complex number $c$, let $\sigma_c\colon\mathbb N\rightarrow\mathbb C$
denote the divisor function defined by
$\sigma_c(n)=\displaystyle{\sum_{d|n}d^c}$ for all $n\in\mathbb N$, and define
$R(c)=\{\sigma_c(n)\in\mathbb C\colon n\in\mathbb N\}$ to be the range of
$\sigma_c$. We study the basic topological properties of the sets $R(c)$. In
particular, we determine the complex numbers $c$ for which $R(c)$ is bounded
and determine the isolated points of the sets $R(c)$. In the third section, we
find those values of $c$ for which $R(c)$ is dense in $\mathbb C$. We also
prove some results and pose several open problems about the closures of the
sets $R(c)$ when these sets are bounded.
