Topological 1-Basedness in Model Theory

• Topological 1-basedness is a model-theoretic notion defined through the use of arbitrary open sets in the forking topology, capturing dependence via local canonical data.
• It underpins dichotomy theorems in hypersimple unidimensional theories by distinguishing between almost p₀-internal unbounded opens and types that are essentially 1-based.
• In t-minimal structures with the independent neighborhood property, topological 1-basedness drives group-existence results, leading to the formation of open, type-definable abelian topological groups.

Topological 1-basedness is a model-theoretic linearity notion that is formulated through topology rather than solely through definability. In one line of work, it appears as a strengthening of essential 1-basedness in simple theories by allowing arbitrary open sets in the forking topology; in another, it is defined in tame topological theories through the interaction of dimension and germs of local definable behavior. In both settings, the point is to detect when dependence is controlled by local canonical data rather than by richer field-like geometry, and this control yields strong structure theorems, including dichotomies for hypersimple unidimensional theories and the existence of open type-definable abelian topological groups in t-minimal structures with the independent neighborhood property (Shami, 2013, Castle et al., 25 Aug 2025).

1. Forking topology and the topological reformulation of 1-basedness

In a complete simple theory $T$, working in a sufficiently saturated model $\mathcal C$, the forking topology $\mathcal T^f_x(A)$ on the Stone space $S_x(A)$ is generated by basic open sets of the form

$U_{\varphi(x,y),b} \;=\; \bigl\{\,p(x)\in S_x(A)\colon \varphi(x,b)\mbox{ forks over }A\bigr\}.$

Equivalently, one regards

$$\{\,a\in\mathcal C^x\colon \varphi(a,b)\text{ forks over }A\}$$

as an open subset of $\mathcal C^{|x|}$, inducing an open on $S_x(A)$. More generally, one may work with a projection-closed family of topologies

$$\Upsilon=\{\,\Tau_x(A)\colon x\text{ a finite variable-tuple},\;A\subset\mathcal C\text{ small}\,\},$$

where each $\Tau_x(A)$ refines the Stone topology, is invariant under automorphisms and renaming of variables, and is closed under products with full Stone spaces and under coordinate projections (Shami, 2013).

Against this background, Shami introduces s-essential 1-basedness via arbitrary opens. If $\mathcal C$0 is projection-closed and $\mathcal C$1 are small, then a complete type $\mathcal C$2 is s-essentially 1-based over $\mathcal C$3 by means of $\mathcal C$4 if for every finite tuple $\mathcal C$5 and every $\mathcal C$6-open set

$\mathcal C$7

such that every $\mathcal C$8 satisfies $\mathcal C$9, the bad locus

$\mathcal T^f_x(A)$0

is nowhere dense in the relative Stone topology on $\mathcal T^f_x(A)$1. The essential change from earlier formulations is that no type-definability assumption is imposed on $\mathcal T^f_x(A)$2: the classical version tests only definable or type-definable $\mathcal T^f_x(A)$3-open sets, while the generalized version tests every $\mathcal T^f_x(A)$4-open set. This moves 1-basedness from a definability-restricted condition to a genuinely topological one.

2. Dichotomy theorems in hypersimple unidimensional theories

The topological reformulation is used to prove a dichotomy for hypersimple theories, where hypersimplicity means simplicity together with elimination of hyperimaginaries, so that canonical bases behave well. Let $\mathcal T^f_x(A)$5 be a partial type over $\mathcal T^f_x(A)$6 of $\mathcal T^f_x(A)$7-rank $\mathcal T^f_x(A)$8. Then one of two alternatives holds: either there is a small parameter set $\mathcal T^f_x(A)$9 and an unbounded $S_x(A)$0-open set

$S_x(A)$1

which is almost $S_x(A)$2-internal, hence in particular of finite $S_x(A)$3-rank; or every complete type internal in $S_x(A)$4 is s-essentially 1-based over $S_x(A)$5 by means of $S_x(A)$6 (Shami, 2013).

The proof proceeds by contraposition. If some type internal in $S_x(A)$7 fails s-essential 1-basedness, then there are a finite tuple $S_x(A)$8, a finite parameter $S_x(A)$9 from $U_{\varphi(x,y),b} \;=\; \bigl\{\,p(x)\in S_x(A)\colon \varphi(x,b)\mbox{ forks over }A\bigr\}.$0, and an open set $U_{\varphi(x,y),b} \;=\; \bigl\{\,p(x)\in S_x(A)\colon \varphi(x,b)\mbox{ forks over }A\bigr\}.$1 such that the bad locus is dense in the Stone topology of $U_{\varphi(x,y),b} \;=\; \bigl\{\,p(x)\in S_x(A)\colon \varphi(x,b)\mbox{ forks over }A\bigr\}.$2. By invariance and projection-closure, one refines to a Stone-open subset $U_{\varphi(x,y),b} \;=\; \bigl\{\,p(x)\in S_x(A)\colon \varphi(x,b)\mbox{ forks over }A\bigr\}.$3 on which each point witnesses algebraic dependence associated with the failure of 1-basedness. One then defines an auxiliary open set

$U_{\varphi(x,y),b} \;=\; \bigl\{\,p(x)\in S_x(A)\colon \varphi(x,b)\mbox{ forks over }A\bigr\}.$4

shows that $U_{\varphi(x,y),b} \;=\; \bigl\{\,p(x)\in S_x(A)\colon \varphi(x,b)\mbox{ forks over }A\bigr\}.$5 is $U_{\varphi(x,y),b} \;=\; \bigl\{\,p(x)\in S_x(A)\colon \varphi(x,b)\mbox{ forks over }A\bigr\}.$6-open, and verifies that it is unbounded and almost $U_{\varphi(x,y),b} \;=\; \bigl\{\,p(x)\in S_x(A)\colon \varphi(x,b)\mbox{ forks over }A\bigr\}.$7-internal.

A central corollary is the unidimensional dichotomy: if $U_{\varphi(x,y),b} \;=\; \bigl\{\,p(x)\in S_x(A)\colon \varphi(x,b)\mbox{ forks over }A\bigr\}.$8 is hypersimple and unidimensional, then either $U_{\varphi(x,y),b} \;=\; \bigl\{\,p(x)\in S_x(A)\colon \varphi(x,b)\mbox{ forks over }A\bigr\}.$9 is s-essentially 1-based or $$\{\,a\in\mathcal C^x\colon \varphi(a,b)\text{ forks over }A\}$$0 is supersimple. In the countable case, under the additional hypothesis

$$\{\,a\in\mathcal C^x\colon \varphi(a,b)\text{ forks over }A\}$$1

the conclusion strengthens: one may replace the unbounded internal open set by an unbounded type-definable $$\{\,a\in\mathcal C^x\colon \varphi(a,b)\text{ forks over }A\}$$2-open set of bounded finite $$\{\,a\in\mathcal C^x\colon \varphi(a,b)\text{ forks over }A\}$$3-rank, and the complementary alternative becomes classical essential 1-basedness for internal, even almost internal, types. When $$\{\,a\in\mathcal C^x\colon \varphi(a,b)\text{ forks over }A\}$$4, this yields a further corollary: for a countable hypersimple theory whose forking-opens over a countable set are themselves countably type-definable, either there is a weakly minimal formula $$\{\,a\in\mathcal C^x\colon \varphi(a,b)\text{ forks over }A\}$$5 which is almost $$\{\,a\in\mathcal C^x\colon \varphi(a,b)\text{ forks over }A\}$$6-internal, or every type internal in $$\{\,a\in\mathcal C^x\colon \varphi(a,b)\text{ forks over }A\}$$7 is essentially 1-based with respect to the forking topology.

The vector-space example clarifies the intended geometry. A possibly uncountable vector space over a fixed division ring is s-essentially 1-based in the forking topology, since all canonical bases of independent tuples lie in their definable closures; correspondingly, the dichotomy predicts no internal open of finite rank.

3. Germs, dimension, and topologically 1-based types in t-minimal structures

A second, more recent notion of topological 1-basedness is formulated for sufficiently saturated t-minimal structures equipped with a definable Hausdorff topology and satisfying the independent neighborhood property (INP). In this setting, Johnson’s dimension theory assigns to each real tuple $$\{\,a\in\mathcal C^x\colon \varphi(a,b)\text{ forks over }A\}$$8 and small parameter set $$\{\,a\in\mathcal C^x\colon \varphi(a,b)\text{ forks over }A\}$$9 a number $\mathcal C^{|x|}$0. One also has a canonical germ of $\mathcal C^{|x|}$1 at $\mathcal C^{|x|}$2,

$\mathcal C^{|x|}$3

which codes the equivalence class of any minimal definable witness $\mathcal C^{|x|}$4 of $\mathcal C^{|x|}$5 over $\mathcal C^{|x|}$6 under agreement on some neighborhood of $\mathcal C^{|x|}$7 (Castle et al., 25 Aug 2025).

If $\mathcal C^{|x|}$8, $\mathcal C^{|x|}$9, and $S_x(A)$0 is small, writing $S_x(A)$1, one says that

$S_x(A)$2

if

$S_x(A)$3

The structure $S_x(A)$4 is topologically 1-based if for every pair of real tuples $S_x(A)$5, the type $S_x(A)$6 is topologically 1-based over $S_x(A)$7.

Assuming INP, there is an equivalent local constancy criterion: $S_x(A)$8 is topologically 1-based over $S_x(A)$9 if and only if the map

$$\Upsilon=\{\,\Tau_x(A)\colon x\text{ a finite variable-tuple},\;A\subset\mathcal C\text{ small}\,\},$$0

is constant on a neighborhood of $$\Upsilon=\{\,\Tau_x(A)\colon x\text{ a finite variable-tuple},\;A\subset\mathcal C\text{ small}\,\},$$1 in $$\Upsilon=\{\,\Tau_x(A)\colon x\text{ a finite variable-tuple},\;A\subset\mathcal C\text{ small}\,\},$$2. This connects the definition directly to the behavior of local definable data under variation of parameters.

Within the t-minimal $$\Upsilon=\{\,\Tau_x(A)\colon x\text{ a finite variable-tuple},\;A\subset\mathcal C\text{ small}\,\},$$3 INP framework, topological 1-basedness is the linear/non-linear dividing line. If $$\Upsilon=\{\,\Tau_x(A)\colon x\text{ a finite variable-tuple},\;A\subset\mathcal C\text{ small}\,\},$$4 is not topologically 1-based, it is called non-linear: there exist points $$\Upsilon=\{\,\Tau_x(A)\colon x\text{ a finite variable-tuple},\;A\subset\mathcal C\text{ small}\,\},$$5 for which fibers in $$\Upsilon=\{\,\Tau_x(A)\colon x\text{ a finite variable-tuple},\;A\subset\mathcal C\text{ small}\,\},$$6 overlap in a field-like way. If $$\Upsilon=\{\,\Tau_x(A)\colon x\text{ a finite variable-tuple},\;A\subset\mathcal C\text{ small}\,\},$$7 is topologically 1-based, Johnson’s generic continuity together with the germ-constant criterion yields a locally linear picture. More precisely, topological 1-basedness is equivalent to the statement that for every real $$\Upsilon=\{\,\Tau_x(A)\colon x\text{ a finite variable-tuple},\;A\subset\mathcal C\text{ small}\,\},$$8, in the projection

$$\Upsilon=\{\,\Tau_x(A)\colon x\text{ a finite variable-tuple},\;A\subset\mathcal C\text{ small}\,\},$$9

any two fibers are either equal or disjoint. The paper identifies this as the exact analog of the weak normality used by Pillay in 1-based stable groups. In the o-minimal case, the notion coincides with the usual linear side of the trichotomy: $\Tau_x(A)$0

4. Group existence from non-triviality and topological 1-basedness

The main structure theorem in the t-minimal setting is a group-existence result. If $\Tau_x(A)$1 is sufficiently saturated, t-minimal, and has INP, and if it is both non-trivial and topologically 1-based, then there is a countable parameter set $\Tau_x(A)$2 and an $\Tau_x(A)$3-type-definable abelian group

$\Tau_x(A)$4

with $\Tau_x(A)$5 open in the ambient topology (Castle et al., 25 Aug 2025).

Here non-triviality means that there are $\Tau_x(A)$6 with $\Tau_x(A)$7 and each of $\Tau_x(A)$8 algebraic over the other two. The proof is organized as a topological groupoid-configuration argument. From non-triviality and topological 1-basedness one extracts tuples $\Tau_x(A)$9 forming a pre-group configuration, while 1-basedness ensures a regular family of homeomorphisms between infinitesimal neighborhoods $\mathcal C$00, $\mathcal C$01, and $\mathcal C$02. This family is interpreted as a regular groupoid spine, and combinatorial group-configuration theorems then produce a type-definable connected group $\mathcal C$03 acting regularly on $\mathcal C$04, so that $\mathcal C$05.

The remaining step is to pass from a local infinitesimal object to an ambient topological group. Generic continuity, Marikova’s local-to-global argument, and compactness and dimension-theoretic arguments are used to upgrade $\mathcal C$06 to a genuinely topological group. The result is an open type-definable topological group $\mathcal C$07 with identity $\mathcal C$08.

A plausible implication is that, in this framework, topological 1-basedness does not merely constrain forking-like dependence; it forces the existence of a canonical linear object living on an open subset of the home sort.

5. Local linearity, local abelianity, and the structure of 1-based groups

Once one has an open type-definable group $\mathcal C$09, the subspace topology from $\mathcal C$10 becomes the natural topology on $\mathcal C$11. Generic continuity implies that on a dense open $\mathcal C$12-type-definable set, multiplication and inversion are continuous. Marikova’s argument then yields, after a small translation, a locally topological group, and further shrinking gives an open type-definable subgroup that is a genuine topological group (Castle et al., 25 Aug 2025).

Inside such a group, the analog of the Hrushovski–Pillay structure theorem takes the following form.

• Locally linear: for each $\mathcal C$13 and each parameter set $\mathcal C$14, the germ $\mathcal C$15 is, up to a translate, a coset of a subgroup of $\mathcal C$16.
• Locally abelian: the infinitesimal subgroup $\mathcal C$17 is abelian, hence there is an open abelian subgroup.
• Few subgroups: any definable family of $\mathcal C$18-dimensional subgroups has constant germ on an open set.

Together these properties imply that a topologically 1-based group looks locally like a module over a discrete ring. The formulation is explicitly presented as the topological analog of the Hrushovski–Pillay classification of 1-based stable groups.

This local structure theorem also clarifies why topological 1-basedness is treated as a linearity notion. The ambient topology determines infinitesimal neighborhoods $\mathcal C$19, and 1-basedness forces these infinitesimal pieces to organize as translated subgroup germs rather than as more complicated non-linear configurations.

6. Examples, scope, and terminological boundaries

The 2025 theory places several familiar tame topological settings on the linear side of the dividing line. The naturally occurring t-minimal theories with INP that are stated to be topologically 1-based are:

• Visceral expansions of fields or groups.
• Dense weakly o-minimal structures, with no exchange assumption.
• C-minimal structures, even without exchange.

In these cases, INP is verified by a uniform dimension argument, and topological 1-basedness follows because the induced definable germs coincide with interval germs in the weakly o-minimal case or ball-germs in the C-minimal case. The stated structural consequence is broad: any non-trivial, topologically 1-based t-minimal $\mathcal C$20 INP theory interprets an infinite open type-definable abelian topological group, extending linear group theorems from the o-minimal and C-minimal settings to a common framework (Castle et al., 25 Aug 2025).

The open directions identified in this line of work are equally structural. They include a full non-linear topological trichotomy, the possibility of removing or replacing INP by new tameness axioms, a global classification of topologically 1-based groups analogous to Pillay’s $\mathcal C$21-structures, and definable compactness or measure-theoretic analogs. In the forking-topological setting, related open questions concern analogous dichotomies without hypersimplicity, the role of other projection-closed topologies such as the $\mathcal C$22-rank topology, and the effective construction of unbounded internal opens witnessing non-1-basedness in concrete examples such as simple groups of higher rank (Shami, 2013).

A potential source of confusion is terminological. In the homotopy-theoretic study of “covering with excess one,” the expression “1-based” is used for a different phenomenon: the covering configuration space deformation-retracts onto a graph, so $\mathcal C$23 for all $\mathcal C$24, and all nontrivial topology lies in dimensions $\mathcal C$25 and $\mathcal C$26 (Wang, 2013). This suggests a sharp distinction between the model-theoretic notion of topological 1-basedness—formulated via canonical bases, germs, dimension, and local constancy—and a separate homotopical usage in which “1-based” refers to one-dimensional homotopy type.