Topology Problems for Challenging Students

[micromass]

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.

 

[lavinia]

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.

- 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.

 

[Bacle]

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

 

[quasar987]

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

 

[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/dp/048668735X/?tag=pfamazon01-20 is a nice source for unusual and interesting problems.

 

[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.

A question asked here recently:
"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:

 

[micromass]

"- 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.

I forgot the word countable.

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

 

[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.

 

[micromass]

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.

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.

 

[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!

 

[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.

 

[Jamma]

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.

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!).

 

[micromass]

- Construct a space that is connected but not path connected

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

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

Excellent question!

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

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.

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

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

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

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!

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

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

Also, the book https://www.amazon.com/dp/048668735X/?tag=pfamazon01-20 is a nice source for unusual and interesting problems.

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.

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

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

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

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.

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

Good one! This goes on my list.

 

[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?

 

[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).

 

[Bacle]

Jamma:

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

 

[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.

 

[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

 

[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.).

 

[micromass]

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.).

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...

 

[Bacle]

Micro: No, I meant telling when the 1-pt compactification is metrizable, as in S^2 for
R^2.

 

[micromass]

Micro: No, I meant telling when the 1-pt compactification is metrizable, as in S^2 for
R^2.

A metrizable space has a metrizable compacification iff it is seperable. A seperable metrizable space can always been seen as subspace of $[0,1]^\mathbb{N}$. So its closure in that space is a metrizable compactification.

The one point compactification of a metrizable space is metrizable iff it is locally compact and separable. This follows from the Urysohn metrization theorem.

 

[Bacle]

I see, because separable+locally-compact is second countable (and 2nd countable spaces
do embed in I^n)?

 

[micromass]

I see, because separable+locally-compact is second countable (and 2nd countable spaces
do embed in I^n)?

Well, seperable is equivalent to second countable in metric spaces.

To prove that the Alexandroff compactification is metrizable, you need the Urysohn metrization theorem, this states that second countable regular spaces are metrizable.

Now, the Alexandroff compactification is seperable, since the original space was. Thus it is also second countable (no need for local compactness here, I think).
The Alexandroff compactification is regular since it is compact Hausdorff. It is Hausdorff since the original space is locally compact.

 

[mathwonk]

have you looked at counterexamples in topology by steen and seebach?

 

[lavinia]

"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).

Try doing this on the closed unit disk.

 

[Jamma]

I'm sorry Lavinia, I don't follow. Are you saying the metric I'm defining won't be a metric? Or that the closed unit disc is such a space? (originally, we wanted a non-compact one, and the metric in this case is bounded by 2).

 

[lavinia]

I'm sorry Lavinia, I don't follow. Are you saying the metric I'm defining won't be a metric? Or that the closed unit disc is such a space? (originally, we wanted a non-compact one, and the metric in this case is bounded by 2).

no. try putting such a metric on the unit disk.

 

[lavinia]

A lot of wild examples come from the theorem that the uniform limit of continuous functions is continuous. what about giving the construction of the sequence that converges to a space filling curve and ask for a proof that the curve is space filling. This also brings up the disturbing question whether space isn't really mult-dimensional but is just a curve wound up in a strange way. One can then start to ask for the meaning of a topological definition of dimension.

One can then take the case of a straight line tilted at an irrational angle on a flat torus and ask for a proof that the line is dense in the torus and yet not space filling.

With the Devil's staircase one could ask for the measure of the set upon which it is constant and also ask questions about its monotonicity and the manner in which it ascends from zero to one on the unit interval.

Using the middle fifths, then sevenths, ... one creates a sequence of staircases, one for each odd number. Does this sequence converge uniformly? What is its limit?

 

[lavinia]

Here is one that I have no idea if it is interesting, hard, easy, profound, or trivial.

Take a basis for the reals over the rationals on the unit interval. What does its subspace topology look like? Is it discrete?

 

[Jamma]

no. try putting such a metric on the unit disk.

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]

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.

The unit disk under the usual metric and your modification, is compact.

 

[Jamma]

The unit disk under the usual metric and your modification, is compact.

Oh, I think I see what you are getting at. I wasn't trying to imply that this procedure makes the space non-compact (the topology will be unaltered). All I'm saying is that for any non-compact complete metric space you can think of, if the metric isn't bounded then you can simply bound it in the way I described.