Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Definable compactness in o-minimal structures

Pablo Andújar Guerrero

TL;DR

The paper proves that in Hausdorff definable topological spaces within an o-minimal framework, multiple natural notions of definable compactness—curve-compactness, filter-compactness, type-compactness, and transversal-compactness (including $(m,n)$-property variants)—are all equivalent. The core approach combines o-minimal VC theory with forking and definable types to translate intersection properties of definable families into the existence of definable limits and transversals, with uniform definability in parameterized families. A key outcome is that definable compactness is definable uniformly in families and, for o-minimal expansions of $\mathbb{R}$, coincides with classical compactness; the paper also provides a non-Hausdorff counterexample showing the necessity of Hausdorffness in certain equivalences. Overall, the results organize a broad landscape of definable compactness notions and pave the way for applying these concepts in various NIP contexts and definable topologies.

Abstract

We characterize the notion of definable compactness for topological spaces definable in o-minimal structures, answering questions of Peterzil and Steinhorn (1999) and Johnson (2018). Specifically, we prove the equivalence of various definitions of definable compactness in the literature, including those in terms of definable curves, definable types, and definable downward directed families of closed sets.

Definable compactness in o-minimal structures

TL;DR

The paper proves that in Hausdorff definable topological spaces within an o-minimal framework, multiple natural notions of definable compactness—curve-compactness, filter-compactness, type-compactness, and transversal-compactness (including -property variants)—are all equivalent. The core approach combines o-minimal VC theory with forking and definable types to translate intersection properties of definable families into the existence of definable limits and transversals, with uniform definability in parameterized families. A key outcome is that definable compactness is definable uniformly in families and, for o-minimal expansions of , coincides with classical compactness; the paper also provides a non-Hausdorff counterexample showing the necessity of Hausdorffness in certain equivalences. Overall, the results organize a broad landscape of definable compactness notions and pave the way for applying these concepts in various NIP contexts and definable topologies.

Abstract

We characterize the notion of definable compactness for topological spaces definable in o-minimal structures, answering questions of Peterzil and Steinhorn (1999) and Johnson (2018). Specifically, we prove the equivalence of various definitions of definable compactness in the literature, including those in terms of definable curves, definable types, and definable downward directed families of closed sets.
Paper Structure (12 sections, 17 theorems, 22 equations, 1 figure)

This paper contains 12 sections, 17 theorems, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conventions
  4. Intersecting families of sets and refinements
  5. Type preliminaries
  6. O-minimal VC theory and forking
  7. $\text{VC}$ theory
  8. Forking, dividing and definable types
  9. O-minimal types
  10. O-minimal definable compactness
  11. Topological preliminaries
  12. Characterizing definable compactness

Key Result

Theorem A

Fix an o-minimal structure $\mathcal{M}=(M, <, \ldots)$. Let $(X,\tau)$ be a definable topological space in $\mathcal{M}$. The following are equivalent. Moreover all the above imply and, if $\tau$ is Hausdorff or $\mathcal{M}$ has definable choice, are equivalent to:

Figures (1)

  • Figure 1: Depicting (in blue) the set $A(x',x",x"',y',y",y"')$.

Theorems & Definitions (46)

  • Theorem A
  • Definition 1.1
  • Lemma 2.1
  • Corollary 3.2
  • proof
  • Corollary 3.4
  • Lemma 3.5
  • proof
  • Corollary 3.7
  • proof
  • ...and 36 more