Lecture 1: Topology (International Winter School on Gravity and Light 2015)

[@josephavant8250]

This professor is EASILY one of the best I've ever seen - every student should be so lucky to study from such an articulate, patient, and clear instructor at some point in their academic career!

[@addemfrench]

I cannot get over how great his presentation is. The ideas are so crystal clear, the notation and board work so pretty and suggestive of the ideas they represent, all of it organized, and even balanced like a painting.

[@elliotnicholson5117]

He's utterly brilliant. :)

[@addemfrench]

Agreed, his lecture is inspiring.

[@boxerpop82]

His lectures are simply beautiful

[@xanthirudha]

this is amazing,i cant believe virtual learning is this promising

[@gentgjonbalaj8359]

Wow! Easily the best lecture I have ever listened to. Thank you!

[@atanunath]

Wow, nobody explained these things so clearly. Brilliant.

[@antoniolewis1016]

This man crafts his lectures from diamonds. He even has board cleaners!

[@jsanch855]

I'm an Electronic Engineer, and I allways want to take a Course where you see Topology, Differential Geometry and Gravity, thenx, by the way, all those asking, what you need to know to understand this course, is just Set Theory and Read and Do Proofs, all the rest is explain.

[@HJ-tf9nw]

Awesome lecture, very clear and well motivated!

[@karimsouidi1]

This is one of the best lectures ever !

[@insignia201]

I have never heard of the term "chaotic topology", I know I have heard it being referred to as a trivial topology or an indiscrete topology. Great lecture nonetheless!

[@lokendrasunar5457]

Great dedicated professor.Very comprehensive lecture .Lucky me.

[@viveknsharma]

Fantastically Well-Planned!

[@TwinDoubleHelix]

These lectures are outstanding. Thank you.

[@pythagorasaurusrex9853]

Great lecture! Wished I had such a competent professor when I studied math. I never really got it, cause lectures were bad. This here is explained easy and one can follow. What I like so much about topology is the fact that you don't need these annoying delta-epsilon-calculations to proove continuity :)

[@xxqq96]

What is the prerequisite for this course?

[@muneer332]

The method of using board is amazing

[@rahnumarahman6227]

Can anybody kindly tell me what literature is being followed here.......the lecture is great but It helps having a literature reference that you can look at.

[@tuneerchakraborty5836]

German Precision.

[@adamlantos2319]

Anybody knows the prerequisites for these videos?

[@adamlantos2319]

***** yeah but you have to know the actual prerequisites before you start searching the internet

[@NSBassler]

+adam landos It is a good idea to have had some basic university level math courses like basic linear algebra and calculus courses. However they are not strictly required. You should also be able to make due with high school level mathematics with some difficulties.

[@ingifreyr]

+adam landos I'm guessing this is a graduate level course, so a BS in physics or mathematics should suffice.

[@cliliv]

I think a basic knowlegde on sets, differential equations and calculus would be enough. These lectures are preparation for a bigger and more richer course on General Relativity, I think. So, if you want to learn more about GR, it would be a great start :)

[@jonabirdd]

The math part cannot be understood without exposure to variational calculus (just the Euler-Lagrange eqns), multivariate calculus, and vector calculus on the level needed to understand maxwell's eqns, for example. The physics part requires exposure to special relativity, and again, some lagrangian mechanics would help.

[@taraspokalchuk7256]

Can such f() be defined that maps M to N and also the chosen topology on M to another topolgy on N?

[@addemfrench]

Sure, we typically define f from M to N and then look for various properties. The most important one is whether the inverse function maps open sets to open sets.

[@taraspokalchuk7256]

addemfrench thanks)

[@theleastcreative]

Did anyone attend this and still have the questions from the tutorials?

[@abstract835]

such clearity=========

[@christerholmsten2060]

Thanks for an excellent lecture, what literature is used during the course if there is any?

[@ErnestYAlumni]

Christer Holmstén So far, I've found that @9400754094) by Norbert Straumann to be the closest in spirit to his lectures, but Schuller's video presentation here is the best and most clear and well-organized (solid?) presentation out there on General Relativity, vs. lecture note, textbook, other media. I really think it's even reference worthy. By the way, I write notes up about these lectures here: drive.google.com/file/d/0B1H1Ygkr4EWJbF9mQXluQVVQTDg/view?usp=sharing and on my wordpress.com blog: ernestyalumni.wordpress.com/2015/05/25/20150524-update-on-gravity_notes-tex-pdf-notes-and-sage-math-implementation-of-lecture-1-tutorial-1-topology-for-the-we-heraeus-international-winter-school-on-gravity-and-light-2015/

[@edithsmith9257]

Totally brill - and his enthusiasm is making millionaires of the blackboard chalk oligarchs.

[@UnforsakenXII]

What mathematics should I know prior to starting this course?

[@NoNamedNobody692]

I would say I have a thorough understanding in both Single and Multivariable Calculus, Differential Equations, Non-Euclidian Geometry and possible a course in Differential Topology. And then comes the physics.... have a grand ol time. Lol.

[@777shadowdragon]

+SFLOVER94 what if the only prerequisites ive taken is youtube? 😂

[@robertwilsoniii2048]

Linear Algebra and Analysis; I recommend Linear Algebra by Levandosky, Principles of Mathematical Analysis by Rudin and Vector Calculus by Marsden and Tromba. You can skip these and go straight to Advanced Calculus by Loomis and Sternberg if you want too, this book will cover the content of this class though.

[@IndianHeathen1982]

Calculus, linear algebra and some exposure to abstract algebra probably.

[@robertwilsoniii2048]

***** Calculus is not necessary at all.

[@leanhdung9848]

Very interesting and inspiring lecture.

[@CGMario]

Thank you for posting this! It's very helpful!

[@BlackEyedGhost0]

I learned: a) The power set (P) of a set (M) is the set which contains all subsets of that set. u∈P(M) u⊆M b) A topology (O) can be defined on a set (M) as a subset of the power set -i) a topology must contain the set (M) and the empty set. ∅,M∈O (∴{∅,M}⊆O⊆P(M)) -ii) the intersection of any two members of a topology must also be a member of the topology. (v∩u)∈O | u,v∈O -iii) the union of any number of members of the topology must also result in a member of the topology. Ui(u)∈O | u∈O (is there any reason it needs to be an indexed set rather than simply v∪u like the previous axiom?) c) Members of a topology are called open sets d) A set is closed if it's compliment (relative in M) is an open set e) A map (f) from set M to set N is continuous if the preimage (with respect to f) of every open set in N is an open set in M (obviously requireing a topology in both). ∀V∈O:preim(V)∈O f) If we have 2 maps (f:M->N and g:N->P) and they're both continuous, then the composition of the two is also continuous. g) A subset (S) of a set with a topology can inherit that topology by taking the intersection of the subset and every element in the topology. Os = {u∩S|u∈O} h) If you restrict a continuous map to a specific subset in the domain and inherit the topology, then the restricted map is still continuous. Nice synopsis for such a long video eh?

[@tuneerchakraborty5836]

+BlackEyedGhost Pretty good. In b) iii) The set is indexed because only a finite number of unions can be taken for the resulting set to be an open set. The finiteness of the index set is pertinent due a technicality regarding some peculiar properties of infinite sets. Apparently all the rules that apply to finite sets don't automatically translate to infinite sets. I could point you to a book if you'd want me to.

[@BlackEyedGhost0]

I'd love to be pointed to a book. And thanks for responding to that. I thought about it for a while and couldn't come up with a reason.

[@teriyakichicken1848]

I just finished pre cal and all this math is sooo daunting, I wonder if it will ever end

[@jaredtramontano5249]

+BlackEyedGhost Of Course the index set is needed! What you've written only says the topology contains finite unions of members... It must contain arbitrary unions

[@fortoday04]

I think d) is the reverse?

[@phillipwong4283]

Very good... Remind me of college days.

[@taraspokalchuk7256]

what is the difference between U(alpha) and U? Is U(alpha) a set of all UєO?

[@alpistein]

The lecture was great, but I got annoyed very quickly over how many curly braces I had to draw in my notes :P

[@CarlosGonzalez-rg6ht]

Wonderful lecture, thank you

[@IgorItkin]

Brilliant lecturer! Just brilliant

I want him to be my lecturer :(, he is amazing!

[@drlangattx3dotnet]

Terrific instructor. Thank you sir.

[@muneer332]

SUPERB SIR SUPERB

[@ADA4750]

Thanks.Very Interesting.

[@TwoonyHorned]

Point set topology is just a matter of language.

[@rahaal2590]

fantastic!

[@RalphDratman]

Superb lecture

[@deleogun1717]

thanks

[@吉田新一-l5w]

he is genius

[@NothingMaster]

Ja Ja --- a great teacher.

[@MrAkashvj96]

Superb

[@MarkMattingleyScott]

Freddy! Moin Moin!

[@lucasdarianschwendlervieir3714]

Ah, the Eintein's view on gravity (as opposed to the Feynman view).