Topologist Sine Curve
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- čas přidán 20. 08. 2024
- Here is one of the most important curves in mathematics. It is an example of a set that is connected, but not path-connected, and is very prominent in topology and analysis.
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![The Most Useful Curve in Mathematics [Logarithms]](http://i.ytimg.com/vi/OjIwCOevUew/mqdefault.jpg)
*thanks for watching*
It's our pleasure
Beautiful, thx for the solution of my biggest concern on the topology class
Today we learned about Topologist Sine Curve. _Joining continuously_ with a previous Dr. Peyam video "Is addition continuous?" where π + e ~ 3 + 3 we could conclude that *Engineer* Sine Curve is a Square Wave between -1 and 1... always rounding to an integer, as below:
*x _ _ rnd(sin(x))*
0 _ _ _ _ _ 0
π/3=1 _ _ 1
π/2=2 _ _ 1
π=3 _ _ _ _ 0
4π/3=4 _ _ -1
3π/2=5 _ _ -1
2π=6 _ _ _ _ 0
A square wave... and that is it.
Whaouh ! The first part is impossible for the moment for me, the second much more. Thank you very much.
Topology is totally new for me, but what a coincidence, today searching for books I found a topology book with this sine curve on the cover
If you do not mind me asking, what book was it?
@@fireemblem2770 Counterexamples in Topology (Dover Books in Mathematics) I just saw it online and I'm not sure if it is the same sine curve, but it reminded me. I do not own the book and don't know if it is good at all
ah yes, the inverted frequency sweep. It would be useful for us control enginerds to sweep infinite frequencies in finite time hahahaha
Particularly useful in Anti-control of chaos...👍
very nice sir It is very easy and good way to describe this question
thats gotta be the funniest proof in maths
There is really nothing more topological than connecting something to itsself without connecting it to itsself
Thanks for such a great content
If I'm not mistaken then, the union of (x,sin(1/x)/x) for all real x /= 0 and (0,y) for all real y would also be connected (and not path connected). That's some funky connection.
I think the proof that F-bar is connected can be made simpler. F-bar is the intersection of all closed sets containing F. B is an open set disjoint from F, so B-complement is a closed set containing F. Therefore F-bar is a subset of B-complement and F-bar is disjoint from B. But B is also a subset of F-bar, so B must be empty.
Say you have a graph built up of little connected lines can you from the line equations turn your graph into a curved graph. The answer is yes you can as you graph one line to the next you can approach the equation for the next line and accend away from the current line formula. The question is what math can you make with little lines that you can curve, what use is it.
Here's some Mathematica code demonstrating what I'm saying.
Plot[Piecewise[{{((0.5 - x)/
0.5) (x/2) + (x/0.5) (0.25 + (x - 0.5) 1.1),
x 0.5 && x 0.75}}] , {x,
0, 1}]
Damn I love topology. It is so weird.
Thanks! It really helps
I dont understund why to consider the union if the two sets.
I wish u could do something for beginners i realy like ur videos but i dont understand anything ...
Check out my college algebra and precalculus playlists
Thanks
Gracias.👀
There is a problem on comb space in algebraic topology book by Hatcher. It is related to the concept of deformation retract in algebraic topology. It is not similar to this. In that problem the main idea is any neighborhood of (0,1) intersection the comb space is not connected. I have to be more precise. But roughly this is the main idea.sin(1/x) is a very good thing for students who want to find counter examples. In this problem the main idea is sin(1/x) oscillates violently as we move nearer to the origin. We cannot control x so that it lies inside a sphere of radius 1/2 with centre (0,1). This is the idea. But in math we have to write a proper proof. Math is unforgiving. unless we write a proper proof mathematicians don’t accept it.
You mean the exercises from Chapter 0?
Can you make a - 6.66hz tone.
Cheers
ist there already a contradiction in the assumption that A,B non empty and open but A u B = F bar? Because the union of two open sets gives again an open set. so this combination of assumptions doesn't work?
A set can be both open and closed
By relative topology
You mean subspace top?
Hey Dr. Peyam, whatever happened to Douglas Ulrich? I’ve tried looking at what he’s been working on but can’t seem to find anything
I’m not really sure, he left the department halfway through 2019 and no word from him any more 😕 He’s fine and alive though, but not sure about the details
It is true for general top spaces, also payam Can I use the same example to show that if the space is path connected than its closure is not path connected necessarily
I want to ask a question. You assume A is a open set and you say that A=F-bar but A is open F-bar by def is a closed set. This seems unreasonable
But I think this is the whole point, the only set in a connected space that is both open and closed is either the empty set or the whole set. Open is not the opposite of closed
@@drpeyam Thank you . I learn a lot from you
Have you made a video on connectedness?
Yes
What does 33 mean?
@@drpeyam It's my school roll number.
What is that?
@@drpeyam It's a number assigned by the school to each student in a class for administrative purposes like maintaining a database.
the fact that you mention it is not your proof is pretty boss
Connected not path connected ? Or it’s closure not
Path connected
Topologist sin curve...
Are you Azari? From azarebiajan
World citizen born in Iran... I suppose
@@alexdemoura9972 bro u wouldn’t be answering with a last name
Like Moura hahahahaha I was asking what city he from in Iran.........
😀Moura: Portuguese Victorious Battle Name like Romans did for Scipius "Africanus" (over Carthage) and Julius Cesar "Germanicus" (over German tribes), in this case for the Moors from Morocco in XIV century over Ottoman empire.😀 I never thought it could me get into trouble.
He lived in many places since young, he speaks 7 or 8 languages 👍
Is it a bad name in farsi?
Iranian people don't like my first name as well, so I shortened to Alex... 😀 I'm kidding I shortened because I also lived in many places and most of the people couldn't pronounce my first name correctly in Portuguese 😀 if have any good suggestions, I could abandon my Christian name my family gave me😦... I also got a Chinese name... but I forgot it...😀
lefties unite!