Exercises on Homeomorphisms and Connectedness

1. Which of the following are homeomorphic? (Informally justify why or why not based on arguments related to content we covered in class)
   
   
   a) S¹= {(x,y) | x² + y ² =1} and {(x,y) | max(|x|,|y|) = 1}, both with the subspace topologies of R².
   
   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) R² and R
   
   d) Grad Prove whether or not the Zariski topology on R² and the finite complement topology on R² are homeomorphic
   
   
2. Prove why each of the following is or is not connected: R_l and R_zar=R_fc
   
   
3. Use ideas of connectedness to informally show that no pair of the following is homeomorphic: (0,1), (0,1], [0,1]