We construct a topology on a given algebraically closed field with a distinguished subfield which is also algebraically closed. This topology is finer than Zariski topology and it captures the sets definable in the pair of algebraically closed fields as above; in the sense that definable sets are exactly the constructible sets in this topology.
On algebraically closed fields with a distinguished subfield
Covers of Multiplicative Groups of Algebraically Closed Fields of Arbitrary Characteristic
On model theory of covers of algebraically closed fields
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