DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 2026-09-07

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.

Euclidean upper half-space and its boundary

Definition

For $n \geq 1$ , put $H^{n} = {(x^{1}, \dots, x^{n}) \in R^{n} : x^{n} \geq 0}$ , with the subspace topology, and put $\partial H^{n} = {x \in H^{n} : x^{n} = 0}$ . Put $H^{0} = R^{0}$ and $\partial H^{0} = \emptyset$ .

Used by

Smooth functions on relatively open half-space sets Definition
Topological manifolds with boundary Definition
The closed half-space as a manifold with boundary Example

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)