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Pablo Andújar Guerrero

Publications and source records attributed to Pablo Andújar Guerrero.

9 recordsLinked to original sources
arXiv2026

NTP2 topological structures

A subset of a topological space is constructible if it is a finite Boolean combination of closed sets. We prove that every NTP$_2$ expansion of $(\mathbb{R},<,+)$ by constructible sets defines only constructible sets, and that definable functions are generically piecewise continuous. The result also holds for all NTP$_2$ expansions of $(\mathbb{Q}_p,+,\cdot)$, and all NTP$_2$ definably complete expansions of ordered groups. In the latter case, the structure is generically locally o-minimal, has definable choice, and carries a well-behaved notion of naive topological dimension. For NIP uniform topological structures, constructibility of definable sets is preserved in the Shelah expansion. We classify strong expansions of $(\mathbb{R},<,+)$ by constructible sets, and obtain results on NTP$_2$ d-minimal structures.

math.LO↗
arXiv2025

Definable separability and second-countability in o-minimal structures

We show that separability and second-countability are first-order properties among topological spaces definable in o-minimal expansions of $(\mathbb{R},<)$. We do so by introducing first-order characterizations -- definable separability and definable second-countability -- which make sense in a wider model-theoretic context. We prove that, within o-minimality, these notions have the desired properties, including their equivalence among definable metric spaces, and conjecture a definable version of Urysohn's Metrization Theorem.

math.LO↗
arXiv2025

Orthogonality and domination in o-minimal expansions of ordered groups

We analyse domination between invariant types in o-minimal expansions of ordered groups, showing that the domination poset decomposes as the direct product of two posets: the domination poset of an o-minimal expansion of a real closed field, and one derived from a linear o-minimal structure. We prove that if the Morley product is well-defined on the former poset, then the same holds for the poset computed in the whole structure. We establish our results by employing the `short closure' pregeometry ($\mathrm{scl}$) in semi-bounded o-minimal structures, showing that types of $\mathrm{scl}$-independent tuples are weakly orthogonal to types of short tuples. As an application we prove that, in an o-minimal expansion of an ordered group, every definable type is domination-equivalent to a product of 1-types. Furthermore, there are precisely two or four classes of definable types up to domination-equivalence, depending on whether a global field is definable or not.

math.LO↗
arXiv2025

Definable compactness in o-minimal structures

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.

math.LO↗
arXiv2023

One-dimensional definable topological spaces in o-minimal structures

We study the properties of topological spaces $(X,τ)$, where $X$ is a definable set in an o-minimal structure and the topology $τ$ on $X$ has a basis that is (uniformly) definable. Examples of such spaces include the canonical euclidean topology on definable sets, definable order topologies, definable quotient spaces and definable metric spaces. We use o-minimality to undertake their study in topological terms, focussing here in particular on spaces of dimension one. We present several results, given in terms of piecewise decompositions and existence of definable embeddings and homeomorphisms, for various classes of spaces that are described in terms of classical separation axioms and definable analogues of properties such as separability, compactness and metrizability. For example, we prove that all Hausdorff one-dimensional definable topologies are piecewise the euclidean, discrete, or upper or lower limit topology; we give a characterization of all one-dimensional, regular, Hausdorff definable topologies in terms of spaces that have a lexicographic ordering or a topology generalizing the Alexandrov double of the euclidean topology; and we show that, if the underlying structure expands an ordered field, then any one-dimensional Hausdorff definable topology that is piecewise euclidean is definably homeomorphic to a euclidean space. As applications of these results, we prove definable versions of several open conjectures from set-theoretic topology, due to Gruenhage and Fremlin, on the existence of a 3-element basis for regular, Hausdorff topologies and on the nature of perfectly normal, compact, Hausdorff spaces; we obtain universality results for some classes of Hausdorff and regular topologies; and we characterize when certain metrizable definable topologies admit a definable metric.

math.LO↗
arXiv2023

Definable $(ω, 2)$-theorem for families with VC-codensity less than $2$

Let $\mathcal{S}$ be a family of sets with VC-codensity less than $2$. We prove that, if $\mathcal{S}$ has the $(ω, 2)$-property (for any infinitely many sets in $\mathcal{S}$, at least $2$ among them intersect), then $\mathcal{S}$ can be partitioned into finitely many subfamilies, each with the finite intersection property. If $\mathcal{S}$ is definable in some first-order structure, then these subfamilies can be chosen definable too. This is a strengthening of the case $q=2$ of the definable $(p,q)$- conjecture in model theory and of the Alon-Kleitman-Matoušek $(p,q)$-theorem in combinatorics.

math.LO↗
arXiv2021

Types, transversals and definable compactness in o-minimal structures

Through careful analysis of types inspired by [AGTW21] we characterize a notion of definable compactness for definable topologies in general o-minimal structures, generalizing results from [PP07] about closed and bounded definable sets in o-minimal expansions of ordered groups. Along the way we prove a parameter version for o-minimal theories of the connection between dividing and definable types known in the more general dp-minimal context [SS14], through an elementary proof that avoids the use of existing forking and VC literature. In particular we show that, if an $A$-definable family of sets has the $(p,q)$-property, for some $p\geq q$ with $q$ large enough, then the family admits a partition into finitely many subfamilies, each of which extends to an $A$-definable type.

math.LO↗