Alphabeta MathCovering Spaces and Lifting
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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✓ 22 results · all verified · 4 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 18 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings
Definition
Maps and isomorphisms of covering spaces over a fixed base
Definition
Lifts of maps, paths, and homotopies through a covering map
Proposition
Covering maps are surjective local homeomorphisms with discrete fibres
Proposition
The cardinality of a covering fibre is locally constant and is constant on a connected base
Theorem
Existence and uniqueness of path lifts through a covering map
Theorem
Existence and uniqueness of homotopy lifts through a covering map
Corollary
The endpoint of a lifted path depends only on its endpoint-fixed homotopy class
Theorem
A covering map induces an injective homomorphism on fundamental groups
Theorem
Two lifts from a connected space that agree at one point agree everywhere
Theorem
Lifting criterion for maps from path-connected locally path-connected spaces
Definition
The pullback of a covering space along a continuous map
Proposition
Covering spaces are stable under restriction, finite products, and pullback
Proposition
A composite of covering maps is a covering when the outer covering is finite-sheeted
Definition
The monodromy right action on a covering fibre and its equivalent left-action convention
Proposition
Monodromy acts by fibre bijections, and its orbits are the intersections of path components with the fibre
Definition
Deck transformations and the deck-transformation group of a covering
Proposition
On a connected covering space, a deck transformation is determined by one point and the deck action is free
Definition
Covering-space actions by disjoint translates of neighbourhoods
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
Proposition
Local path-connectedness lifts and descends along covering maps
Definition
Semilocally simply connected spaces with explicit basepoint convention
Definition
Universal covering spaces
Theorem
A space admitting a universal covering is semilocally simply connected
Definition
The based path-class model and basic sets for a universal cover
Lemma
For a nonempty path-connected locally path-connected semilocally simply connected space, the path-class projection is a covering map
Theorem
Every nonempty path-connected locally path-connected semilocally simply connected space has a universal cover
Theorem
For a path-connected locally path-connected base, a universal cover maps uniquely over the base to every connected covering, and any two universal covers are uniquely isomorphic
Corollary
A connected covering of a locally path-connected simply connected space is one-sheeted and trivial
Theorem
For a nonempty path-connected total space, a covering fibre is in bijection with the right cosets of the induced fundamental-group subgroup
Theorem
For a finite-sheeted covering, the total space is compact exactly when the base is compact
Theorem