Alphabeta MathHow 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.
Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings
Definition
A covering map is a continuous surjection such that every has an open neighbourhood for which is a disjoint union of open sets , called sheets, and each restriction is a homeomorphism (Continuity of a map of topological spaces at a point and globally, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological, The disjoint union (coproduct) with the final topology of the canonical injections: a set is open exactly when each of its traces is). Such a is evenly covered, and is the fibre over . A covering is trivial when it is isomorphic over to a product projection with discrete.
Depends on
- Continuity of a map of topological spaces at a point and globally
- Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological
- 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
- The disjoint union (coproduct) $\bigsqcup_i X_i$ with the final topology of the canonical injections: a set is open exactly when each of its traces is
Used by
- A connected covering of a locally path-connected simply connected space is one-sheeted and trivial Corollary
- Lifts of maps, paths, and homotopies through a covering map Definition
- Maps and isomorphisms of covering spaces over a fixed base Definition
- The pullback of a covering space along a continuous map Definition
- Universal covering spaces Definition
- Pulling a covering back to an evenly covered open set gives a trivial covering Example
- The maps [x]↦[mx] on ℝ/ℤ are m-sheeted coverings for m≥1 Example
- Trivial coverings are products with a discrete fibre Example
- For a nonempty path-connected locally path-connected semilocally simply connected space, the path-class projection is a covering map Lemma
- A composite of covering maps is a covering when the outer covering is finite-sheeted Proposition
- Covering maps are surjective local homeomorphisms with discrete fibres Proposition
- Covering spaces are stable under restriction, finite products, and pullback Proposition
- Local path-connectedness lifts and descends along covering maps Proposition
- The cardinality of a covering fibre is locally constant and is constant on a connected base Proposition
- For a finite-sheeted covering, the total space is compact exactly when the base is compact Theorem
- The orbit map of a covering-space action is a covering, with the acting group equal to the deck group when the total space is path-connected Theorem
- Two lifts from a connected space that agree at one point agree everywhere Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 25 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Allen Hatcher, Algebraic Topology, §1.3 (standard reference, not scraped)
- J. Peter May, A Concise Course in Algebraic Topology, Ch. 3 (standard reference, not scraped)
- Marco Gualtieri, MAT1300 Week 4 Term 2, §1.6 (standard reference, not scraped)