Enumeration degrees and non-metrizable topology
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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 ω-power of the Sierpiński 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.
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