  The enumeration degrees of sets of natural numbers can be identified with the
degrees of difficulty of enumerating neighborhood bases of points in a
universal second-countable $T_0$-space (e.g. the $\omega$-power of the
Sierpi\'nski space). Hence, every represented second-countable $T_0$-space
determines a collection of enumeration degrees. For instance, Cantor space
captures the total degrees, and the Hilbert cube captures the continuous
degrees by definition. Based on these observations, we utilize general topology
(particularly non-metrizable topology) to establish a classification theory of
enumeration degrees of sets of natural numbers.
