math250-extprobs

# math250-extprobs - Extra Problems for Math 250A Some of...

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Extra Problems for Math 250A Some of the (parts of) exercises below repeat or are special cases of results prove in class. If X is a Hausdor/ topological space then a covering space for X is a pair ( °; Y ) of Y a Hausdor/ topological space and ° : Y ! X a surjective continuous map such that if p 2 X then there exists U an open neighborhood of p in X such that U is evenly covered by ° . This means that ° ° 1 ( U ) = [ ° 2 I U ° with, U ° open, U ° \ U ± = ; if ± 6 = ² and ° j U ° is a homeomorphism of U ° onto U . The cardinality of I is called the order of the covering if j I j = r then the covering is called an r -fold cover. 1. Prove that if M is a C 1 manifold and ( °; Y ) is a covering of M as a topological space then there exists an atlas f ( U ² ; ° ² ) g ² 2 J for M such that each U ² is evenly covered by ° . 2. Let M; °; Y be as in 1. and let f ( U ² ; ° ² ) g ² 2 J be as in 1. Let ° ° 1 ( U ² ) = [ ° 2 I ± U °² be the disjoint union as in the de°nition of evenly covered. De°ne ° °² = ° ° ° ° j U °± . Show that

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