Oral test 1 on material including Problem Set 2 and 3. Review definitions, examples, and reasons why statements are true or false from the problem sets and solution, the WebCT quiz, and class notes.

This test will be closed to notes.

For example, I might ask you to define a metric ball in a space with the discrete topology, give an example of a set with two different bases that generate the same topology on the set and to explain why (open circles and open squares generate the same topology on R2 and we explained why in class using the basis comparison of topology ideas), discuss the product topology examples from class notes...

Review especially (ie many of the questions will be very similar to those below):

• Definitions of a metric ball
• Definition of a topology
• Various definitions of an open set in a topology
• Definition of a basis for a topology
• A metric ball in the discrete metric topology and an explanation of why
• Metric balls in R and R^2 and an explanation of why
• Examples of open sets in the cofinite topology on R (and sets that are not open) and an explanation of why
• Examples of open sets in the lower limit topology on R (and sets that are not open) and an explanation of why
• Example of a subbasis which is not a basis and the topology it generates and an explanation
• The product topology on RlxRl and the subspace topology it induces on lines of different slopes and an explanation of why
• Two topologies so that the union is not a topology and an explanation of why
• Examples of topologies (without using the indiscrete or discrete topologies) so that the first topology is strictly contained in the second and an explantion of why
• Explanations of why certain topologies are not contained in others (example why isn't the lower limit topology contained in the standard topology?)
• Examples of two different bases that generate the same topology and and an explanation of why ...

In each case, I will be looking for clarity and depth, and a clear demonstration that you understand "why". For the following Thursday, I will ask you to reflect on your oral presentations via a self-evaluation. So think about what are aspects of your presentations that went especially well? How about aspects that could use improvement? I will also ask you to give yourself a grade.

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. If you can make some connections, relate it to big picture ideas or metaphors, that is good too.