I'm making this thread because in a few weeks I'll be starting with teaching a topology course. I think I did pretty well last time I teached it, but I want to do some new things. The problem with the system in our country is that we hardly assign problems that students should solve. Students are expected to solve problems on their own (which they usually don't do). The results of this system can be witnessed at the final exam where we notice that many students just didn't invest much time in practicing the concepts.(adsbygoogle = window.adsbygoogle || []).push({});

Anyway, this time I really want to challenge the students, and I want to do this by giving a rather difficult problem at the end of the week which they should ponder on and solve (with the help of books if necessary). The difficult problems shouldn't be too difficult though and should have an easy solution (once you know it). Furthermore, I want the problems to be rather fun and curious results which might even challenge their intuition on topics.

Some problems I've had in mind:

- [itex]\mathbb{R}[/itex] cannot be written as the disjoint union of closed intervals.

- HallsOfIvy's problem on connected sets: https://www.physicsforums.com/showpost.php?p=3494673&postcount=9

- Prove that the pointswise convergence does not come from a metric

- Take a countable number of lines away from [itex]\mathbb{R}^3[/itex]. Is the resulting space connected?

I'm making this thread to ask if anybody has any challenging topology problems that might be suitable.

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# Topology problems

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