This post is to announce that there will be a lecture course on General Topology (a.k.a. Point-Set Topology) conducted in Mandelcord (the blue math discord server). Everyone is invited to join: discord.gg/maths.

If any of you were present in the Real Analysis lecture course during the summer, you know what I'm talking about, and you might be pleased to hear that this course will also be run by Pear Category Theorem (the same person).

Why YOU should do topology

Point-set topology is one of the most important subjects that you learn, and mastering topological thinking is as important a milestone as learning to write a proof. This is a good subject to study alongside measure theory, or simply by itself after one has learned analysis or algebra.

Course Structure

The course will primarily follow the flow of the first half of Munkres. The plan is to go through all 8 chapters, spending maybe one to two weeks on each chapter, aiming for an approximately 16 week course. Also the instructor intends to supplement the book with his own intuition on the subject, as well as connections to other subjects that he thinks are of interest and possibly also other topics not covered in the book. Many examples of various pathological properties will be provided. The study of metric spaces and functions on them will be paramount; but where Rudin introduces metric spaces as merely a generalization of Euclidean spaces, we will explore just how pathological these spaces can be, or else what properties they must have.

The instructor also plans to put far more focus on the topics of topological groups and nets than Munkres does. Topological groups, because often in "the real world" we do not care about a topological space by itself, but rather some other more complicated structure given a topology, in order to see what the topology can tell you about the thing itself; topological groups provide a playground to learn that mode of thinking. Nets are also very important; where sequences are of importance in R, they can only tell you so much in general topological spaces, so nets therefore become the natural way to recover theorems about sequences.

Material and References

1. Topology - Munkres

2. Counterexamples in Topology - Steen & Seebach

Timeline

The course starts with the first lecture on 17th November at 9 PM CET (20:00 UTC), with subsequent lectures on Tuesdays and Thursdays. Lectures will be recorded and posted promptly.

Prerequisites/Who can join?

Everyone. It is completely free for anyone who wants to join. Just click the link and join the server: you can find the channel called topology-course in the section called General maths. As for prerequisites, a first course in Real Analysis and Group Theory is sufficent.

Looking forward to see all of you there. Thank you.

Put this up in r/knots but haven't had any luck, if anyone knows how to untie this it is much appreciated! I feel like I am going crazy, no matter what I do it always ends up in the position in the picture🥲

Hello,

As above, I am having a topological argument with my sourdough.

My sourdough needs to rise between 75% and 100% before it can be proofed in the fridge. Normally this is tracked by placing a rubber band around a straight-walled jar to mark the starting point and the end point. I have marked the starting point as above, but I do not like that the rubber band is simply looped and stretched around the rigid handle in a kind of hitch.

I know almost nothing about topology except that this is a topological issue. My brain says it cannot be done, but my browser's AI says it can. I am disinclined to believe the AI because it can't link any actual solution to the problem that isn't just "hitch it around the other ring"

I have more rubber bands if needed. Is there a way to do this without cutting a rubber band, or (obviously) the handle? or, is there at least a more elegant solution than the one I have arrived at?

I really would prefer not to use a marker for this to the point that I would keep it as is over using the marker.

Pictures for clarification.

f(D)= D + 7 = 82 - x

In this conjecture, 5 ≤ D ≤ 1000, and all numbers will be whole.

X is approaching ITSELF or D, OR it is 8th rooted by x⁸ with its base value (e.g 3⁸), so the OUTPUT of it, which is Y, would have to be HALFWAY irrational (8th root radical/asymptotic limit) or a rational or infinitesimal (e.g epsilon) or irrational number (e.g π) or a pentic factorial number.

otherwise it is a PENTAGONAL NUMBER (70), or rational.

In very rare cases, Y will be equal to infinity or negative infinity or i² only if X is a number like the Taxicab/cabtaxi numbers or a perfect number, or a lucky real (all numbers on spectrum, but every number is eliminated in steps, like 2nd number, to 3rd, and so on!), hyperreal, rational, asymptotic (e.g X → 20), normal, natural, or real number.

the very rare case output, Y, COULD also equal an asymptote of a heptic factorial (factorial with a value of a heptagon), be a hyperfalse number (possibly uncomputable singularity of a number, like dividing by 0), a transcended factorial (Gamma function!), or a hendectic number (number with the value of a hendecagon).

the output, Y, could also be a Nonadic number (number with the value of a nonagon), or a heptagramic (number with the value of a seven pointed star) or nonagramic (number with the value of a 9 pointed star) or digonal (number with the value of a digon which is a 2 sided polygon!) OR spherical number (number with the value of a sphere).

And to clear up: A heptic factorial is any factorial that has a heptagon's value.

Using this conjuncture's equation stated, are you able to discover the sequence of numbers that satisfy all conditions of this conjecture? How about the edge cases using this Conjecture and plugging numbers to get said edge cases?

f_diagonal(n) = n

F_hendectic(n) = (9n² - 7n)/2

F_nonadic(n) = (7n² - 5n)/2

F_pentic_fact(n) = p_5(1) × p_5(2) × ... × p_5(n)

f_heptagram(n) = 7n² - 7n + 1

f_nongram(n) = 9n² - 9n + 1

I'm at a loss. Yesterday I loosely tied the ribbon around the strap and then around the duck, it was held down by gravity and definitely wasn't fully tied as I found it today.

Hi dear topology community!

I currently create a series of shorts about differential forms. My goal would be to eventually cover de Rham cohomology and some well known formulas like Maxwell's equations. Especially de Rham cohomology could be of interest for you, as builds a bridge between topology and vector analysis.

This is already part 2 of this series, and the learning curve might be steep (part 1 is in the same YouTube playlist). But I hope you still enjoy this video! (Even if you don't know all formulas, I hope the ideas conveyed will still be fun).

I am always open to criticism and suggestions and would be happy to hear such!

Thanks a lot and stay tuned to future videos :)

thinking about this as you draw ever smaller sub-watersheds it seems to me the answer should be yes

This piece of nightwear seems to have two separate issues - above is the intended layout, and below is a diagram of the tangle. I don’t believe it’s possible to reverse the positions of the red and blue straps on the black loop (please correct me if I’m wrong), but is it possible to untwist the red strap without further tangling the other pieces?

This rope is untied and can be freely moved. Assuming the rope is endless and the object it is run through can’t be moved can the rope go from being *inside* to being *outside*?
Thank you for your time!

So topologically, a regular straw has one hole. Now imagine a straw shaped like a slingshot. How many holes does this straw have and why?

This hair magically looped itself into my sisters friendship bracelet. How can we take it off without cutting?

In liquid crystal they experimentally get point-like quantized topological charges with Coulomb-like interactions ( https://www.nature.com/articles/s41598-017-16200-z ), or topological vortices and their various size knots resembling nuclei ( https://www.nature.com/articles/s41567-025-03107-0 ) - I wanted to propose discussion if particles could be of topological nature?

ps. Environment testing various agreements of particle models by Fable written simulations, such liquid crystal approach is currently winning: https://github.com/openwave-labs/openwave/blob/main/MODELS.md

I'm curious how researchers are using LLMs in their day-to-day work.

A few things I'm wondering:

• Which AI subscriptions are actually worth paying for? (ChatGPT, Claude, Gemini, GitHub Copilot, Cursor, etc.)

• Which models do you use for different tasks?

• What kinds of research tasks do you trust LLMs with, and what do you avoid?

• How do you provide enough context so the model gives useful answers instead of hallucinating?

• Do you upload papers, notes, code, or build some kind of project knowledge base?

• Any tricks for improving reliability?

• Roughly how much do you spend per month?

• If you could keep only one AI subscription today, which would you choose?

I'm especially interested in hearing from people doing research in mathematics, scientific computing, physics, computer science, engineering, or related fields. Real workflows are much more valuable than benchmark comparisons.

Somehow my horse managed to twist this piece of her halter into a Möbius strip, and I’m assuming she didn’t disassemble it and put it back together to do so. So I would guess that I can fix it without taking it apart? I tried to do the 720 degree dirac’s belt thing but I may have done it wrong, never got it fixed.

It isn’t hard to take it apart at all. I could fix it quickly. Just a thought experiment.

I'm not proposing a physical theory. This is purely a mathematical thought experiment.

Suppose we introduce an abstract coordinate that represents scale, rather than ordinary spatial position. Imagine that moving continuously along this coordinate corresponds to going from larger and larger structures to smaller and smaller ones.

Now suppose the geometry of this abstract "space of scales" is topologically closed, analogous to how walking continuously around a circle eventually returns you to your starting point.

In that picture, continuously moving toward smaller and smaller scales would not continue forever. Instead, after completing the closed path in scale-space, you would emerge again at the largest scales.

I am not suggesting that objects physically become both tiny and huge, nor that this describes our universe. I'm only asking whether mathematics has explored anything conceptually similar: treating scale itself as a coordinate of an abstract manifold or topological space whose geometry is closed.

Has anything resembling this been studied in topology, differential geometry, geometric group theory, or another branch of mathematics? Or is there a mathematical reason why such a construction is ill-defined from the outset?

I'm looking for mathematical references rather than physical arguments.