### Exercises on Homeomorphisms and Connectedness

**Which of the following are homeomorphic?** (Informally justify
why or why not based on arguments related to content we covered in class)

a) S^{1}= {(x,y) | x^{2} + y ^{2} =1}
and {(x,y) | max(|x|,|y|) = 1}, both with the subspace topologies
of R^{2}.

b) R with the standard topology and R_cf with the finite complement
topology.

c) and from the first day of class.

d) [1,2) and {0}U(1,2)

e) R^{2} and R

d) __Grad__ Prove whether or not
the Zariski topology on R^{2} and the finite
complement topology on R^{2} are homeomorphic

- Prove why each of the following is or is not connected:
R
_{l}
and R_{zar}=R_{fc}

- Use ideas of connectedness to informally show that no pair of
the following is homeomorphic: (0,1), (0,1], [0,1]