TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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

Statement

Let $B$ be path-connected and locally path-connected. After basepoints over the same point are fixed, a universal cover of $B$ admits a unique based continuous map over $B$ to every connected covering of $B$ ; in particular any two universal covers of $B$ are uniquely isomorphic over $B$ .

The map to a connected covering is asserted here only as a continuous map over the base. The classical stronger form, that this map is itself a covering map, needs a surjectivity argument and evenly covered neighbourhoods for it, and is not established on this page.

Facts & Assumptions

Given: The objects, hypotheses, and choice principles stated above.

[F1]

A universal covering space of $B$ is a covering map $p : \tilde{B} \to B$ whose total space $\tilde{B}$ is simply connected (def-covering-map-and-evenly-covered-neighbourhoods, def-simply-connected). (Universal covering spaces).

[F2]

Let $Y$ be path-connected and locally path-connected, let $f : (Y, y_{0}) \to (B, b_{0})$ be based, and let $p : (E, e_{0}) \to (B, b_{0})$ be a covering. A based lift $\tilde{f} : (Y, y_{0}) \to (E, e_{0})$ exists if and only if $f_{*} π_{1} (Y, y_{0}) \subseteq p_{*} π_{1} (E, e_{0})$ ; when it exists it is unique. (Lifting criterion for maps from path-connected locally path-connected spaces).

[F3]

Let $Y$ be connected and let $f, g : Y \to E$ be lifts through the same covering of the same map $Y \to B$ . If $f (y_{0}) = g (y_{0})$ for some $y_{0} \in Y$ , then $f = g$ . (Two lifts from a connected space that agree at one point agree everywhere).

[F4]

For a covering $p : E \to B$ , the total space $E$ is locally path-connected if and only if the base $B$ is locally path-connected. (Local path-connectedness lifts and descends along covering maps).

Proof

technique · direct

1.1

Let the base be path-connected and locally path-connected and fix points over a common basepoint.

givenF4F2F1

2.1

By [F1] the total space of a universal cover is simply connected, so its fundamental group is trivial and the subgroup condition of [F2] holds for any target covering. [F2] therefore supplies a based lift to any connected covering, and asserts that lift to be unique. What [F2] delivers is a continuous map over the base; it carries no covering-map conclusion, and no later step supplies one.

step 1.1F1F2F3

3.1

Applying this in both directions between two universal covers and using uniqueness of lifts makes the composites identities.

step 2.1F3F1F4

4.1

The preceding construction and implications establish the assertion.

step 3.1∎

Depends on

Used by

For a path-connected locally path-connected semilocally simply connected base, the deck group of a universal cover is isomorphic to the fundamental group Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 36 results over 12 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)