DefinitionDefinition: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Ambient closure and intrinsic boundary of a hypersurface

Definition

For a subset $S \subset M$ , write ${\overline{S}}^{M}$ for its closure in $M$ : a point lies in it precisely when every open neighborhood in $M$ meets $S$ . Give ${\overline{S}}^{M}$ its subspace topology. The ambient boundary of any subset $A \subset M$ is $\partial_{M} A = {\overline{A}}^{M} ∖ {int}_{M} A$ , where ${int}_{M} A$ is the union of the open subsets of $M$ contained in $A$ .

A topological three-manifold with boundary is a Hausdorff, second-countable space locally homeomorphic to relatively open subsets of ${(y^{1}, y^{2}, y^{3}) : y^{3} \geq 0}$ . Its manifold boundary is defined by the boundary-chart convention of the supplier. For hypersurface completion data, require ${\overline{S}}^{M}$ to be such a manifold, $S$ to be its manifold interior, and $B : = {\overline{S}}^{M} ∖ S$ to be its manifold boundary and a topological two-manifold without boundary. In particular $B$ is a boundary of the completed three-manifold, whereas $\partial_{M} S$ is an ambient boundary; the symbols are not interchangeable. No smoothness of $B$ in $M$ is required.

Definition references

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Topological manifolds with and without boundary

Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace

Facts & Assumptions

[F1]

The smooth manifold has its manifold topology. Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

[F2]

Manifolds with boundary and their intrinsic boundary have the specified chart-based meaning. Topological manifolds with and without boundary

[F3]

For a subset $A$ of a topological space $X$ , its subspace topology consists of the sets $U \cap A$ with $U$ open in $X$ ; this also defines relatively open subsets. Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace

Justification

**Given:**The objects and specified data in the Definition.

Goal: Verify that the displayed conditions define unambiguous objects and predicates on these data.

1.1 The neighborhood predicate determines a unique closure; the union of open subsets determines a unique interior. Their set difference therefore determines the ambient boundary, without selecting charts. [F1]

2.1 The manifold-with-boundary conditions are predicates on the specified subspace topology; requiring that its intrinsic boundary equal $B$ is an additional condition, not a consequence of the closure operation. Thus the completion data distinguish both boundary notions even when only topological boundary charts exist. [F1, F2, F3, step 1.1] ∎

Depends on

Used by · 3

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.