Oral test 3 on material up through and including Problem Set 6. Review definitions, examples, and reasons why statements are true or false from the recent two problem sets and solutions, the WebCT quiz, and class notes.

This test will be closed to notes and help and comments can only come from Dr. Sarah.

Review especially the definitions that I listed for the material from Oral test 1 and Oral test 2 as it is likely that some of these definitions will reappear on this test. In addition, make sure that you also know

• Numerous definitions of continuous
• Definition of 1-1
• Definition of onto
• Definition of homeomorphism
• Definition of connected
• Definition of a separation
• Definition of path connected

In addition, be sure to know numerous examples for each of the following (or an explanation of why numerous examples are not possible):

• Two spaces that are subspaces of R and are homeomorphic.
• Two spaces that are subspaces of R and are not homeomorphic.
• Two spaces that are subspaces of R2 and are homeomorphic.
• Two spaces that are subspaces of R2 and are not homeomorphic.
• A space that is connected but not path connected
• A space that is path connected but not connected
• A space that is neighter connected nor path connected
• A space that is both connected and path connected

For each example, you need to demonstrate why it satisfies the criterion.

In addition, know whether the following are homeomorphic and why or why not.

• R with the standard topology and R_cf with the finite complement topology.
• S1 and [0,1)
• The Zariski topology on R2 and the finite complement topology on R2

In each case, I will be looking for clarity and depth, and a clear demonstration that you understand "why". Don't be afraid of repetition - when you get up to the board, restate your question, explain the relevant background material, answer your question, and then review what you did and explain again how it relates to the original question. Make some connections, and relate it to big picture ideas or metaphors too. I am looking for:
Clarity
Depth
Your demonstration of an understanding of relevant definitions and the problem and its correct solution without help from others
Eye Contact
Included enough repetition
Included connections and big picture ideas or metaphors