HW1 120909 - Differential Geometry Homework 1- 15.05.08...

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Unformatted text preview: Differential Geometry Homework 1- 15.05.08 Hava Shabtai, ID 043039619, Department of Mathematics, University of Haifa Email: —hshabtai@study.haifa.ac.il— 1. Any subset of R n endowed with the subspace topology is Hausdorff becase R n is Hausdorff. Taking any pair of distinct points p,q ∈ X are distinct points in R n which is Haus- dorff and so there exist disjoint open subsets (of R n ) U ⊆ R n containing p and V ⊆ R n containing q hence U X := U ∩ X and V X := V ∩ X are disjoint open subsets (of X this time) U X ⊆ X containing p and V X ⊆ X containing q . From similar argument, any subset of R n endowed with the subspace topology is second countable becase R n is second countable. Taking B to be the set of all open balls with rational radii and rational center point is a countable basis for R n . Define B X := { B ∩ X | B ∈ B} and we received countable basis for X . 2. We will presenet two explicit atlas, the first for S 2 the second for T 2 . (a) We know that the sphere S 2 ⊆ R 3 is a submanifold of diention 2 and therefor a smooth manifold. For each index i = 1 , 2 , 3 let U + i denote the subset of S 2 where the i th coordinate is positive: U + i = ( x 1 ,x 2 ,x 3 ) ∈ S 2 : x i > ....
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This note was uploaded on 09/16/2009 for the course MATH 106723 taught by Professor Michaelpolyak during the Spring '09 term at Technion.

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HW1 120909 - Differential Geometry Homework 1- 15.05.08...

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