Dimension in T.T.T Structures

TL;DR

This paper explores two notions of dimension in products of one-dimensional topologically totally transcendental structures, proving their equivalence in certain cases and analyzing the density of sets.

Contribution

It introduces and compares topological and algebraic dimensions in t.t.t structures, establishing their equivalence in ta-saturated cases and analyzing set density.

Findings

Topological and algebraic dimensions are equivalent in ta-saturated 1D t.t.t structures.

Dense sets in these structures are comeager.

The paper provides a framework for understanding dimensions in t.t.t structures.

Abstract

In this paper we consider two types of dimension that can be defined for products of one-dimensional topologically totally transcendental (t.t.t) structures. The first is topological and considers the interior of projections of the set onto lower dimensional products. The second one is based on algebraic dependence. We show that these definitions are equivalent for \omega -saturated one-dimensional t.t.t structures. We also prove that sets which are dense in products of these structures are comeager.