This discussion focuses on innovative approaches to teaching topology, specifically by introducing challenging problems to engage students. The instructor aims to enhance problem-solving skills by assigning thought-provoking tasks such as proving that \(\mathbb{R}\) cannot be expressed as the disjoint union of closed intervals and exploring connectedness in modified spaces. Participants contribute additional problem ideas, emphasizing the importance of intuition and curiosity in learning topology. The conversation highlights the need for effective problem assignments to improve student performance and understanding in topology courses.
PREREQUISITESMathematics educators, topology students, and anyone interested in enhancing their understanding of advanced topology concepts through problem-solving.
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.