Minimal and Topologically Faithful Internal Models

• Minimal and topologically faithful internal models are canonical definable group structures constructed in t-minimal structures with INP, capturing linear and locally modular behavior.
• They leverage an interplay of infinitary topology and model-theoretic stability to derive type-definable abelian topological groups that are open, minimal, and internal to their ambient settings.
• These models generalize o-minimal and C-minimal group-existence theorems, unifying linear phenomena across visceral, weakly o-minimal, and C-minimal theories.

Minimal and topologically faithful internal models are canonical definable group structures constructed within highly saturated, t-minimal structures equipped with the independent neighborhood property (INP) and satisfying topological 1-basedness. These models serve as geometric representations of linear, locally modular behavior generalizing o-minimal and C-minimal group-existence theorems to a broad class of "tame" topological theories, encompassing all visceral, weakly o-minimal, and C-minimal theories irrespective of exchange properties. Their construction is underpinned by a precise correspondence between infinitary topology and model-theoretic stability, culminating in type-definable abelian topological groups that are minimal, topologically faithful, and internal to the ambient structure (Castle et al., 25 Aug 2025).

1. t-Minimality and the Independent Neighborhood Property

A structure $\mathcal M$ is called t-minimal if it possesses a uniformly $\emptyset$-definable Hausdorff topology on its domain $M$ such that for every definable $X \subseteq M$,

$$|X| = \infty \iff \operatorname{int}(X) \neq \emptyset.$$

Equivalently, every infinite definable set contains a definable non-empty open subset. The independent neighborhood property (INP) further stipulates that for each tuple $a \in M^n$, small parameter set $A \subseteq M^{eq}$, and definable neighborhood $U \ni a$, there exists a definable neighborhood $V \subseteq U$ and parameter $t$ ensuring

$\emptyset$0

capturing dimension invariance under the introduction of suitable parameters. This property is fundamental for isolating precisely controlled neighborhoods relevant for the ensuing group construction (Castle et al., 25 Aug 2025).

2. Topological 1-Basedness and Germ Codes

Topological 1-basedness is a generalization of model-theoretic 1-basedness employing topological and dimensional data. For tuples $\emptyset$1, $\emptyset$2 and a small parameter set $\emptyset$3, write $\emptyset$4 for the germ at $\emptyset$5 of the definable set of realizations of $\emptyset$6. The type $\emptyset$7 is topologically 1-based over $\emptyset$8 if

$\emptyset$9

i.e., $M$0's dimension given $M$1 coincides with its dimension given the germ $M$2. Equivalently, by Lemma 5.2,

$M$3

is locally constant in a neighborhood of $M$4. The structure $M$5 itself is topologically 1-based if this property holds for all real tuples $M$6 over $M$7. This notion allows a precise linear/non-linear dichotomy among t-minimal structures with INP, mirroring linear phenomena in dimension and topology (Castle et al., 25 Aug 2025).

3. Existence and Construction of Internal Topological Groups

Under the hypotheses that $M$8 is highly saturated, t-minimal, has INP, is non-trivial (i.e., has parameters $M$9 with $X \subseteq M$0, each algebraic over the other two), and is topologically 1-based, one obtains a type-definable abelian group $X \subseteq M$1 satisfying:

• $X \subseteq M$2 is open in $X \subseteq M$3,
• $X \subseteq M$4 is a topological group (the topology on $X \subseteq M$5 is inherited as a subspace topology),
• $X \subseteq M$6 is locally linear, with each infinitesimal neighborhood $X \subseteq M$7 forming a (left or right) coset of a subgroup of $X \subseteq M$8.

Initial construction starts with a type-definable group $X \subseteq M$9 on an infinitesimal neighborhood $$|X| = \infty \iff \operatorname{int}(X) \neq \emptyset.$$0 ($$|X| = \infty \iff \operatorname{int}(X) \neq \emptyset.$$1); a countable type-definable $$|X| = \infty \iff \operatorname{int}(X) \neq \emptyset.$$2 open in $$|X| = \infty \iff \operatorname{int}(X) \neq \emptyset.$$3 is found using shrinking definable neighborhoods. By generic continuity and Marikova's local-to-global argument, $$|X| = \infty \iff \operatorname{int}(X) \neq \emptyset.$$4 further acquires a genuine topological group structure (Proposition 10.5), wherein group axioms and continuity extend from a dense open set to the entire open group (Castle et al., 25 Aug 2025).

4. Structural Properties: Local Linearity and Abelianity

The central structural theorems mirror stable group theory and extend the Hrushovski–Pillay classification into the topological domain. If $$|X| = \infty \iff \operatorname{int}(X) \neq \emptyset.$$5 is topologically 1-based, the following hold:

• Local linearity: For any $$|X| = \infty \iff \operatorname{int}(X) \neq \emptyset.$$6 and parameter set $$|X| = \infty \iff \operatorname{int}(X) \neq \emptyset.$$7, the infinitesimal leaf $$|X| = \infty \iff \operatorname{int}(X) \neq \emptyset.$$8 is a coset of a subgroup of $$|X| = \infty \iff \operatorname{int}(X) \neq \emptyset.$$9 (Lemma 11.3).
• Local abelianity: There exists an open type-definable abelian subgroup $a \in M^n$0 (Theorem 11.8).
• Few subgroups: Any type-definable family of subgroups of fixed dimension has constant infinitesimal germ (Theorem 11.7).

These properties express that the group structure around infinitesimal neighborhoods is strictly controlled, preventing non-linear or pathological behavior. In notation: $a \in M^n$1 signifying coset-wise constancy of product maps in small neighborhoods (Castle et al., 25 Aug 2025).

5. Minimality and Topological Faithfulness

The group $a \in M^n$2 constructed in this setting is minimal in the sense that it has t-minimal dimension and admits no proper infinite definable subgroups near the identity. It is topologically faithful because the topology on $a \in M^n$3 is precisely the one inherited from $a \in M^n$4, ensuring that definable open sets in $a \in M^n$5 are open in $a \in M^n$6. The group is internal, being defined by type-definable sets indexed over countable or small parameter sets. Collectively, these properties ensure $a \in M^n$7 functions as a canonical model internal to $a \in M^n$8 capturing the linear/modular aspects of the ambient topology (Castle et al., 25 Aug 2025).

6. Connections and Unification with O-Minimal and C-Minimal Theories

The theory of minimal and topologically faithful internal models synthesizes prior group-existence results in o-minimal and C-minimal settings as particular cases. Specifically, any o-minimal or C-minimal structure automatically satisfies t-minimality and INP, and their canonical topological groups also satisfy the minimality and faithfulness criteria enumerated above. The current framework further encompasses "visceral" and weakly o-minimal theories, even if the exchange property fails, thus providing a unified blueprint for the emergence of internal group structures in a model-theoretically tame topological environment (Castle et al., 25 Aug 2025).