- Prove
*from the definitions*: The continuous image of a connected set is connected. - Prove
*from the definitions*: If*X*is compact and*Y*is Hausdorff and*f*:*X*-->*Y*is continuous and one-to-one, then*f*is a homeomorphism onto its image. - Show by an example that the hypothesis ``
*X*compact'' is necessary.

- Prove
- Prove that any uncountable subset of the plane R^2
has a limit point.
- Suppose
*A*: R^2 --> Z is any function from the plane to the integers. For each*n*let*S*_*n*be the set of points*x*such that*A*(*x*) =*n*. Prove that for at least one number*n*the set*S*_*n*has a limit point in R^2.

- Prove that any uncountable subset of the plane R^2
has a limit point.
- What topological properties of the real line are used in
proving the following two statements?
- The ``Intermediate Value Theorem:'' If [
*a*,*b*] is a closed interval, and*f*a continuous function defined on [*a*,*b*], then for any*c*in the interval [*f*(*a*),*f*(*b*)] there is an*x*in [*a*,*b*] with*f*(*x*) =*c*. - If
*f*is a continuous real-valued function defined on a closed interval [*a*,*b*], then there exists*x*in [*a*,*b*] such that*f*(*x*) > =*f*(*y*) for any*y*in [*a*,*b*]

- The ``Intermediate Value Theorem:'' If [