Problem Sheet B

This Problem Sheet is based on Lectures 2 and 3.

• Problem 1 is a linear analogue of the notion of a smooth atlas on a manifold.
• Problem 2 gives an entirely new way to think about the tangent space, this time via equivalence classes of charts and vectors. Hint: You should use Problem 1 when solving this!
• Problems 3 and 4 show that the tangent space to a vector space is canonically isomorphic to the vector space itself. (NB: These problems won’t make sense until after Monday’s lecture.)
• Problem 5 is an example of a locally Euclidean space which is not Hausdorff.
• Problem 6 is an example of a paracompact Hausdorff space which is locally Euclidean (of positive dimension) but has uncountably many components.

Remark

The “standard” example of a Hausdorff locally Euclidean space that is not paracompact is the so-called “Long line”. This unfortunately is a little too difficult to set as an exercise.