The discussion revolves around the search for challenging topology problems suitable for a course aimed at engaging students more effectively. Participants share various problem ideas that could stimulate students' thinking and understanding of topology concepts, while also addressing the challenges of the current educational system regarding problem-solving expectations.
Participants generally agree on the need for challenging problems but express differing opinions on specific problem formulations and their implications. Some disagreements arise regarding the properties of certain sets and the validity of proposed problems.
Some problems may depend on specific definitions or assumptions that are not universally agreed upon. For example, the discussion about the disjoint union of closed intervals and the properties of connected sets highlights varying interpretations and understandings of topology concepts.
Educators and students interested in topology, particularly those looking for engaging problems to enhance understanding and stimulate discussion in a classroom setting.
lavinia said:no. try putting such a metric on the unit disk.
Jamma said:I'm sorry, I really don't understand.
What would be wrong with my metric on the unit disk? Maybe you misunderstood what I was saying- I define a new metric in terms of the older one. For any metric d, we can construct another one (called d^, say) which is defined as being equal to d except when d evaluates to a number larger than 1, in which case, you set d^ to return the value 1.
lavinia said:The unit disk under the usual metric and your modification, is compact.