# 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* - 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.

Comments and questions?