# Covering Maps : Let q: X->Y and r:Y->Z be covering maps; let p=(r(q(x))). Show if r^(-1)(z) is finite for each z in Z, p is a covering map.

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Let q: X->Y and r:Y->Z be covering maps; let p=(r(q(x))). Show if r^(-1)(z) is finite for each z in Z, p is a covering map.

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#### Solution Preview

r o q is continuous and onto, as both these properties are preserved by

compositions. So only the "evenly covered bit" needs to be checked.

So fix z in Z. It has a neighbourhood U_z such that U_z is evenly covered

by finitely many open sets V_1,...,V_n and r, where r^-1(z) = {y_1,...,y_n}

and each y_i is in V_i. Each V_i is mapped ...

#### Solution Summary

Covering maps are investigated. The solution is detailed and well presented.

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