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Covering space

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Summary

A map p ⁣:Eβ†’Bp\colon E\to B such that every point of BB has an open neighborhood UU for which pβˆ’1(U)p^{-1}(U) is a disjoint union of open sheets, each mapped homeomorphically onto UU by pp.

Record metadata

Carrier(s)

Data

Axioms / constraints

Notes

A covering is a map p ⁣:Eβ†’Bp\colon E\to B together with topologies on both EE and BB satisfying the evenly-covered-neighborhood condition; it is not structure on one topological space alone.

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Fundamental group Ο€1(X,x0)\pi_1(X,x_0)Covering space

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This is an authored directed relation from the source endpoint to the target endpoint.

Authored explanation

For a path-connected, locally path-connected, semilocally simply connected space, connected covering spaces correspond to conjugacy classes of subgroups of the fundamental group.

How to interpret this relation type

Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.

Relation sources

topological spaces + covering map(construction junction)Covering space

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This is an authored directed relation from the source endpoint to the target endpoint.

Authored explanation

Give EE and BB topologies and a continuous surjection p ⁣:Eβ†’Bp\colon E\to B; require every point of BB to have an evenly covered open neighborhood.

How to interpret this relation type

Feed several structures into a construction junction and impose compatibility between them.

Relation sources

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No direct relations are authored in this direction.