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

As part of the world-wide celebrations of the 100th anniversary of Einstein's theory of general relativity and the International Year of Light 2015, the Scientific Organizing Committee makes available the central 24 lectures by Frederic P Schuller.

Titled "A thorough introduction to the theory of general relativity", the lectures introduce the mathematical and physical foundations of the theory in 24 self-contained lectures. The material is developed step by step from first principles and aims at an audience ranging from ambitious undergraduate students to beginning PhD students in mathematics and physics.

Satellite Lectures (see other videos on this channel) by Bernard F Schutz (Gravitational Waves), Domenico Giulini (Canonical Formulation of Gravity), Marcus C Werner (Gravitational Lensing) and Valeria Pettorino (Cosmic Microwave Background) expand on the topics of this central lecture course and take students to the research frontier.

Access to further material on www.gravity-and-light.org/lectures and www.gravity-and-light.org/tutorials

such clearity=========

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.

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

This is one of the best lectures ever !

Terrific instructor. Thank you sir.

Fantastically Well-Planned!

he is genius

What is the prerequisite for this course?

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

Very interesting and inspiring lecture.

Superb lecture

Great dedicated professor.Very comprehensive lecture .Lucky me.

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!

For anyone interested in the problem sheets: gravity-and-light.herokuapp.com/tutorials

Awesome lecture, very clear and well motivated!

The method of using board is amazing

SUPERB SIR SUPERB

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.

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.

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

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

@addemfrench thanks)

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.

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

at 23:18, the equation should be ∀p∈M since Bᵣ is only defined for p∈ℝⁿ.

He did write that.

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

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 :)

If anyone is interested in other lectures by him www.video.uni-erlangen.de/indizes/person/alpha/s/48/id/829 two of them are in english, the other two in german.

Does anyone know where to get the problem sets for this series? It looks like the original site is gone.

I cannot see the tutorial part. could you give me any advice ,sir

ok, I have found them :)

Are there any solutions?

+Dustin Kippers gravity-and-light.herokuapp.com/tutorials

What mathematics should I know prior to starting this course?

i'd like to know how you would talk about smoothness and integration on manifolds without calculus

Basic knowledge of differentiation etc. is needed when talking about smooth manifolds later on in the course. Lecture 4 I think.

@IndianHeathen1982 Calculus is not necessary at all.

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

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.

Point set topology is just a matter of language.

fantastic!

Anybody knows the prerequisites for these videos?

All you need is Linear Algebra and Analysis.

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.

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 :)

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

+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.

Superb

Wow, nobody explained these things so clearly. Brilliant.

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

German Precision.

Wonderful lecture, thank you

Very good... Remind me of college days.

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

His lectures are simply beautiful

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!

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

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?

@The Destroyer A set is closed if and only if its complement is open, so either direction works.

I think d) is the reverse?

+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

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

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.

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

Brilliant lecturer! Just brilliant

thanks

These lectures are outstanding. Thank you.

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

Ja Ja --- a great teacher.

He's utterly brilliant. :)

Agreed, his lecture is inspiring.

Thanks.Very Interesting.

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

@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/

Freddy! Moin Moin!