# Approximation theorems for spaces of localities

## TL;DR

This paper extends classical approximation theorems to infinite sets of valuations and orderings, removing the need for pairwise independence, thus broadening the scope of approximation in field theory.

## Contribution

It introduces new approximation theorems for infinite sets of valuations and orderings without the independence condition, generalizing previous results.

## Key findings

- Established approximation theorems for infinite valuation sets
- Removed the independence condition in approximation theorems
- Broadened the applicability of approximation methods in field theory

## Abstract

The classical Artin--Whaples approximation theorem allows to simultaneously approximate finitely many different elements of a field with respect to finitely many pairwise inequivalent absolute values. Several variants and generalizations exist, for example for finitely many (Krull) valuations, where one usually requires that these are independent, i.e. induce different topologies on the field. Ribenboim proved a generalization for finitely many valuations where the condition of independence is relaxed for a natural compatibility condition, and Ershov proved a statement about simultaneously approximating finitely many different elements with respect to finitely many possibly infinite sets of pairwise independent valuations. We prove approximation theorems for infinite sets of valuations and orderings without requiring pairwise independence.