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I. Let M be a topological space with a topology T. Recall that a function f : M M is called continuous if the preimage by f of any open set in T is in T. Precisely, for A E T, define (A) = € M I f(x) € A). Then f is continuous if for any A € T, f-((4) E T. A similar definition holds for functions f between two distinct topological spaces. We will explore how the choice of topology affects the set of continuous functions. (a) Consider the topology To with basis given by (x), for all x € M. Namely, any subset of M obtained as union of singletons or finite intersections are in T. Characterize the continuous functions on M for this topology. (b) Consider the topology To = (M,0) and the real-line the Borel topology. Characterize the set of continuous functions from M to R for the topologies chosen. 2. A commonly encountered operation in modelling mechanical systems is the cross-product x of vectors in R³: given v = (v1 V2 v3) and w = (w1 W2 w3). we have y X w = - W2U3 U3W1 V1W3 U1W2 - 201212) This operation turns the vector space R³ into an algebra, that is a vector space equipped with a vector- valued "product operation" between vectors. A matrix Lie algebra 0 is a vector space of matrices equipped with the Lie bracket ]: for A, B € g, [A, B] == AB-BA. Two algebras (L1, X1) and (L2, x2) are isomorphic if there exists a linear, invertible map $ LL L2 which "agrees" with the algebra operation, i.c. (AX1B)=(4) X2 ((B). Show that the Lie algebra (so(3), [, 1 and (R³, x) are isomorphic as algebras. Define the set st(2) of 2 x 2 real, traceless matrices. Show that this space is a matrix Lie algebra (i.e. show that it is closed under commutation of matrices). Is it isomorphic to so (3)? 3. Can you show that SO(2) is a manifold? How would you show that SO(n) is a manifold? 6. The Hairy Ball Theorem states that there does not exist continuous, non-zero vector fields on the sphere 52. Using this fact, can you show that there does not exist a global basis for TS2? Namely, there does not exist a pair of vector fields U1 (x), so that = TxS2. The above fact is often referred to as saying that the tangent bundle of s² is not trivial. Can you show that the tangent bundle of 51 is trivial? 7. Consider the space S = [0, ) x [0, 2x) and let (0.4) € S. Consider the map F S S2 (0,4) - (sin A cos Q. sin 0 sin $, cos @) and the vector field v = $80 on S. What is the pushforward of y by F? I

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