Show that a connected metric space having more than one point is uncountable.

Let be a continuous map. Show there exists a point of such that .

Give an example of a subset of having uncountable many connected components. Can such a subset be open? Closed?

A standard theorem states that a continuous real valued function on a compact set is bounded. Prove the converse: If is a subset of and if every continuous real valued function on is bounded, then is compact.

An accurate map of California is spread out flat on a table in Evans Hall, in Berkeley. Prove that there is exactly one point on the map lying directly over the point it represents.

Let be the Sierpinski space (that is the space with the topology determined by letting , , and be the open sets), and prove that is the only space (up to homeomorphism) such that if is any other topological space then continuous functions from to are in bijection with open sets of .

Manifold Theory

Let be the real line with differentiable structure given by the maximal atlas of the chart . Let be the real line with differentiable structure given by the maximal atlas of the chart and . Show that these two differentiable structures are distinct, and that there is diffeomorphism between and .

Let the additive group act on on the right by where is an integer. Show that the orbit space is a smooth manifold.

Let be the map . Let . Compute .

Let be a Lie group with multiplication map and inverse map . Prove that the differential of at the identity is addition of tangent vectors. Prove that the differential of at the identity is negation of tangent vectors.

Let , , , and be the standard coordinates on . Is the solution set of in a smooth manifold assuming the subset is given the subspace topology?

Let the unit sphere in be defined by the equation . If , prove that is tangent to if and only if .

Show that a smooth map from a compact manifold to has a critical point.

Prove that if is a compact manifold, then an injective immersion is an embedding.

Prove two vector fields and on a manifold are equal if and only if for every function on , we have .

Let be the vector field on . For each point , find the maximal integral curve of starting at .

If and are fuctions and and are vector fields on a manifold , show that .