# The Dyck and the Preiss separation uniformly

## TL;DR

This paper provides a constructive, uniform proof for two classical separation theorems involving analytic sets, using separation trees, and derives corollaries related to effective bi-analyticity.

## Contribution

It introduces separation trees to give constructive, uniform versions of Dyck and Preiss separation theorems, extending their classical results.

## Key findings

- Constructive proof of separation theorems using separation trees
- Uniform versions of Dyck and Preiss separation theorems
- Corollaries related to effective bi-analyticity and Souslin-Kleene Theorem

## Abstract

We are concerned with two separation theorems about analytic sets by Dyck and Preiss, the former involves the positively-defined subsets of the Cantor space and the latter the Borel-convex subsets of finite dimensional Banach spaces. We show by introducing the corresponding separation trees that both of these results admit a constructive proof. This enables us to give the uniform version of the separation theorems, and derive as corollaries the results, which are analogous to the fundamental fact "HYP is effectively bi-analytic" provided by the Souslin-Kleene Theorem.