Definable Continuous Injective Maps

• Definable continuous injective maps are functions on open, definable sets that are continuous and one-to-one within o-minimal or locally o-minimal structures.
• Generalizations of the invariance of domain theorem demonstrate that these maps preserve openness, particularly when the Jacobian determinant maintains a constant sign.
• They connect analytic criteria with topological properties, leading to applications in definable manifold theory and covering space constructions through controlled dimension theory.

Definable continuous injective maps are a fundamental class of functions within the model-theoretically tamed settings of o-minimal and locally o-minimal structures, with significant implications for geometric topology, real algebraic geometry, and definable manifold theory. These are functions $f: U \to M^n$—with $U \subseteq M^n$ an open, definable set in a suitable expansion $\M$ of an ordered group or field—such that $f$ is definable, continuous, and injective. Central results include generalizations of the classical invariance of domain theorem, explicit criteria for openness and global embedding, and connections to dimension theory and definable topology.

1. Fundamental Definitions and Context

In a fixed o-minimal expansion of the real field or, more generally, a definably complete locally o-minimal expansion of an ordered group, a definable continuous injective map is a function $f: U \to M^n$, where:

• $U\subseteq M^n$ is definable and open.
• $f$ is definable (relative to the language of $\M$).
• $f$ is continuous in the order (product) topology.
• $f$ is injective.

Associated concepts include the fiber $f^{-1}(y)=\{x\in U : f(x)=y\}$, the differentiability locus $D_f$ where $f$ is $C^1$, the Jacobian determinant $J_f(x):=\det Df(x)$ at differentiable points, and the branch set $$B_f=\{x\in U : f \text{ is not a local homeomorphism at } x\}$$. The o-minimal notion of dimension, denoted $\dim X$, is used throughout for definable sets, with $\dim X<k$ implying measure zero and key connectivity consequences (Dinh et al., 2021).

2. Invariance of Domain for Definable Continuous Injections

The generalization of invariance of domain to the definable setting asserts that every definable continuous injective map from a definable open subset of $M^n$ to $M^n$ is open, provided the structure is definably complete and locally o-minimal. Formally, if $\M=(M,<,+,0,\dots)$ is definably complete and locally o-minimal, and $U\subseteq M^n$ is definable open, then any definable continuous injective $f: U \to M^n$ is open: for all open $W\subseteq U$, $f(W)$ is open in $M^n$ (Fujita, 26 Jan 2026).

The proof exploits the definable dimension function and properties such as: the removal of a closed definable set of dimension $<n-1$ from an open box leaves it definably connected; the boundary $\partial W$ of a non-dense open set $W$ in an open box has dimension exactly $n-1$. Dimension-theoretic arguments, plus injectivity and continuity, guarantee preservation of openness.

3. Equivalent Criteria for Openness and Embedding

Within o-minimal expansions of the real field, precise characterizations of when a definable continuous injective $f: \Omega \to \mathbb R^n$ is open (and hence a topological embedding) are available (Dinh et al., 2021). For $\Omega$ open, connected, and definable, the following are equivalent for injective and continuous definable maps:

1. $f$ is open (and thus a topological embedding of $\Omega$ onto the open set $f(\Omega)$).
2. $J_f(x)$ does not change sign on the differentiability locus $D_f$.
3. $\dim B_f \le n-2$, i.e., the branch set is of codimension at least $2$.

If $D_f$ is dense in $\Omega$ and $J_f$ has no sign change, then $f$ is a homeomorphism onto its image. The result unifies the topology of definable mappings with analytic criteria, leveraging o-minimal dimension and stratification theorems.

4. Definable Inverse and Global Invertibility Theorems

Definable analogues of the classical inverse and global invertibility theorems have been extended to continuous definable mappings, encompassing injectivity and global homeomorphism criteria (Truong et al., 2021):

• Local Inverses: At regular points (where the lower limit of the co-norm of the Jacobian is positive), any continuous definable $f: U \to \mathbb R^n$ restricts locally to a homeomorphism onto its image. These regular points form an open set, and $f$ is locally bi-Lipschitz.
• Global Homeomorphism Criteria: Hadamard-type conditions, such as the absence of asymptotic critical values or an integral lower bound for the co-norm of the Jacobian, guarantee that a definable local homeomorphism is actually a global homeomorphism.
• Necessary Conditions: Properness plus sign-definite Jacobian (nonnegativity almost everywhere) yield surjectivity and “quasi-interiority” of the image, but do not alone guarantee openness; specific examples demonstrate necessity of local homeomorphism assumptions.

These theorems rely on cell decomposition, the definable curve selection lemma, and dimension bounds on the critical and branch loci.

5. Dimension Theory and Control of Exceptional Sets

A structural feature of o-minimal and locally o-minimal settings is the tameness of dimension: for definable sets $X\subseteq \R^n$, sets of codimension at least $1$ are non-separating and of measure zero. The branch locus $B_f$ of non-local homeomorphisms, and the locus where the Jacobian vanishes, are both definable and typically of small dimension compared to the ambient space (Dinh et al., 2021, Truong et al., 2021). This dimensional sparseness underlies the effectiveness of topological arguments and the extension of degree theory to definable settings.

6. Applications to Definable Manifolds and Further Developments

The invariance-of-domain theorem in the definable context directly extends to locally definable $d$-manifolds: any locally injective definable continuous map between such manifolds is open, thus yielding a robust topological theory of definable manifolds in locally o-minimal, definably complete settings (Fujita, 26 Jan 2026). Other applications include definable covering space theory and the construction of definable fundamental groups.

Comprehensive references for proofs and technical details include Đinh & Phạm ("On definable open continuous mappings") (Dinh et al., 2021), the recent invariance of domain result in locally o-minimal structures (Fujita, 26 Jan 2026), and the extension of classical analysis and topological degree arguments to definable continuous mappings by Kurdyka and collaborators (Truong et al., 2021). The tame topological properties assured by local o-minimality and definable completeness are crucial; without these, pathological behaviors not compatible with the invariant results may arise.

7. Summary Table: Equivalent Criteria for Openness—Injective Definable Maps on Open Sets

Criterion	Description	Source
Openness	$f$ is open $\Leftrightarrow$ topological embedding	(Dinh et al., 2021)
Jacobian Sign	$J_f(x)$ has constant sign on $D_f$	(Dinh et al., 2021)
Branch Set Codimension	$\dim B_f \le n-2$	(Dinh et al., 2021)
Local Inverse at Regular Points	$V_f(x)>0\implies$ local homeomorphism	(Truong et al., 2021)

Each of these conditions, when appropriately formulated in the definable setting, provides a rigorous and unified foundation for the study and application of definable continuous injective mappings.