Topology problems

1. Sep 13, 2011

micromass

Staff Emeritus
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.

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:
- $\mathbb{R}$ 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 $\mathbb{R}^3$. Is the resulting space connected?

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

2. Sep 13, 2011

lavinia

- Construct a space that is connected but not path connected

- Find a complete metric space whose metric is bounded - less that or equal to 1 - but which is not compact

- The Cantor set has measure zero and is complete and totally disconnected. What about the set obtained by removing all of the middle fifths?

- Show that the Euler characteristic of a polygon or a polygonal solid such as a prism is independent of the triangulation.

3. Sep 14, 2011

Bacle

Why don't you use a textbook like Munkres, or like Hatcher (free) for the class?

4. Sep 14, 2011

quasar987

Find a space in which the connected components are not clopen.

5. Sep 14, 2011

Petek

I like the following problem because it illustrates that non-Hausdorff spaces can be ill-behaved:

In the finite complement topology on $\mathbb{R}$, to what point or points does the sequence $x_n = \frac{1}{n}$ converge?

Also, the book https://www.amazon.com/Counterexamp...735X/ref=sr_1_1?ie=UTF8&qid=1316014554&sr=8-1 is a nice source for unusual and interesting problems.

Last edited by a moderator: May 5, 2017
6. Sep 14, 2011

Jamma

"- R cannot be written as the disjoint union of closed intervals."

It can!:

{[x,x]| x \in R}

I suppose you need to say that the intervals contain more than one element.

"Can a continuous bijective map (X,T)->(X,T) fail to be a homeomorphism?"
I think is a nice question.

Oh, and: "I think I did pretty well last time I teached it"

"taught it"- Sorry, being a grammar nazi :tongue2:

7. Sep 14, 2011

micromass

Staff Emeritus
I forgot the word countable.

Anyway, these are some very good questions. The students will certainly enjoy themselves!!

8. Sep 14, 2011

Jamma

For hallsofivy's question, can't we take something weird, like all (x,y) such that x,y are irrational for out set P and Q the complement? (with the exception of the corners, of course). That seems to me like it'd work.

9. Sep 14, 2011

micromass

Staff Emeritus
Doesn't work. The irrationals are totally disconnected. Hence your set is not connected. (if I understood you correctly).

The idea is to do something nifty with the topologists sine function.

10. Sep 14, 2011

Jamma

"Find a space in which the connected components are not clopen."

This is a great question- I hadn't thought about it before and have always just assumed that all connected components are clopen. Good questions make you examine your intuition more carefully!

11. Sep 14, 2011

Bacle

Every compact metric space is the continuous image of the Cantor set/space.

Find examples of spaces where sequential continuity does not imply continuity.

12. Sep 14, 2011

Jamma

Ah ok. I thought that maybe it wouldn't be when you take the two dimensional counterpart. To be honest, I didn't really think about it all that hard.

(EDIT: Thinking it about it more hardly, I'm still finding it hard to convince myself that such sets are disconnected in 2D, although I'm probably missing something very obvious. EDIT2: Nope, I've convinced myself now that it is disconnected! Note though, that the set P which consists of all the complement of all rational pairs is actually path connected!).

Last edited: Sep 14, 2011
13. Sep 14, 2011

micromass

Staff Emeritus
A good one, but it's already in the syllabus :tongue2:

Excellent question!!

Another very cool question. Too bad that they did not yet see Lebesgue measure at that point. But I can ask about compactness, connectedness and whether it's perfect.

Nice one!! The course doesn't cover much algebraic topology though

Wow, this is an incredibly good question!! Why didn't I think of this. I might even use the opportunity to talk about quasicomponents and stuff!

Good one. I'm planning a lesson on non-Hausdorff topologies, so this will certainly pop up!

Yes, it's a very cool book. In fact, I'm going to assign everybody of the class a topology from this book which they have to discuss. I hope they learn to appreciate the weirdness of counterexamples.

It is a nice question! I'll use it for sure.

Hmm, I'm having mixed feelings about this. The theorem is super-important and is used in many places. But its proof is somewhat too long. Maybe I'll give the proof and ask them to work out the details. It's a very nice result, though.

Good one! This goes on my list.

Last edited by a moderator: May 5, 2017
14. Sep 14, 2011

Jamma

I think I figured out what you mean about the topologists sine function.

It reminds me of another, similar question. Can you find 3 regions of the plane which all share the same boundary?

15. Sep 14, 2011

Jamma

"Find a complete metric space whose metric is bounded - less that or equal to 1"

For this, can't you just find a complete, non compact metric space and bound the metric? I.e. define the metric d^(x,y)=d(x,y) for d(x,y) < 1, and d^(x,y)=1 otherwise.

Of course, you need to show that d^ is still a metric (not hard).

16. Sep 14, 2011

Bacle

Jamma:

The metric d/(1+d) is equivalent to the metric d, for any metric.

17. Sep 14, 2011

Petek

If A is a subset of a topological space, then at most 14 sets can be constructed from A by complementation and closure. There is a subset of the real numbers (with the usual topology) from which 14 different sets can be so constructed.

18. Sep 15, 2011

Jamma

"Jamma:

The metric d/(1+d) is equivalent to the metric d, for any metric."

Ah, ok. Either works then, just "chopping off" the higher values works fine too, although that metric is a bit prettier in that it doesn't just look like a bodge job

19. Sep 15, 2011

Bacle

Jamma: I guess it may also be possible if X has a (metrizable) compactification, tho
I don't know enough to tell when that is possible.

And, Micromass: there is another one for you: every locally compact+Hausdorff
space has a 1-pt compactification, and S^2 is the 1-pt compactification of
R^2 (Comapctification in this case is Hausdorff, but I don't know the general
rule for when it is.).

20. Sep 15, 2011

micromass

Staff Emeritus
In fact: every space has a one-point compactification. But it's only Hausdorff if the original space is locally compact and Hausdorff. This is an iff even.
It's a good exercise...