TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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.

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.

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

1.1givenF4F2F1

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

2.1step 1.1F1F2F3

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.

3.1step 2.1F3F1F4

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

4.1step 3.1∎

The preceding construction and implications establish the assertion.

Depends on

Used by

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on 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)