# Topological Study of Pairs of Algebraically Closed Fields

## TL;DR

This paper introduces a new topology on algebraically closed fields with a distinguished subfield, capturing definable sets as constructible sets and refining the Zariski topology.

## Contribution

It constructs a topology on pairs of algebraically closed fields that precisely characterizes definable sets as constructible sets, enhancing the understanding of their geometric structure.

## Key findings

- The topology is finer than the Zariski topology.
- Definable sets correspond exactly to constructible sets in this topology.
- Provides a new geometric perspective on pairs of algebraically closed fields.

## Abstract

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.