classification of covering spaces


Let X be a connectedPlanetmathPlanetmath, locally path connected and semilocally simply connected space. Assume furthermore that X has a basepoint *.

A covering p:E→X is called based if E is endowed with a basepoint e and p⁢(e)=*. Two based coverings pi:Ei→X, i=1,2 are called equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath if there is a basepoint preserving equivalence T:E1→E2 that covers the identityPlanetmathPlanetmath, i.e. T is a homeomorphism and the following diagram commutes

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