A text that can accompany this Course on "Geometry for theoretical physics" Dr. Fredric Schuller has uploaded a course on youtube (here) that is intended to cover the geometry which is used the study of theoretical physics. His treatment is mathematically rigorous (I've watched the first $6$ lectures or so). He has mentioned a "textbook" many times during the lectures but he never explicitly mentions  the name of that textbook. I've sent him a message asking for the name of the textbook but a response was never received. So, I wonder if you can recommend any textbook that can go along with those lectures. 
The text need not  contain logic nor set theory nor general topology since those are familiar topics for me, so it's not an issue at all if the text does not cover those. The rest is what matters for me. 
Here are the topics covered in the lectures: 

Introduction/Logic of propositions and predicates- 01 
Axioms of set Theory - Lec 02 
Classification of sets - Lec 03 
Topological spaces - construction and purpose - Lec 04 
Topological spaces - some heavily used invariants - Lec 05 
Topological manifolds and manifold bundles- Lec 06 
Differentiable structures definition and classification - Lec 07 
Tensor space theory I: over a field - Lec 08 
Differential structures: the pivotal concept of tangent vector spaces - Lec 09 - 
Construction of the tangent bundle - Lec 10 
Tensor space theory II: over a ring - Lec 11 
Grassmann algebra and deRham cohomology - Lec 12 
Lie groups and their Lie algebras - Lec 13 
Classification of Lie algebras and Dynkin diagrams - Lec 14 
The Lie group SL(2,C) and its Lie algebra sl(2,C) - lec 15 
Dynkin diagrams from Lie algebras, and vice versa - Lec 16 
Representation theory of Lie groups and Lie algebras - Lec 17 
Reconstruction of a Lie group from its algebra - Lec 18 
Principal fibre bundles - Lec 19 
Associated fibre bundles - Lec 20 
Conncections and connection 1-forms - Lec 21 
Local representations of a connection on the base manifold: Yang-Mills fields - Lec 22
Parallel transport - Lec 23 
Curvature and torsion on principal bundles - Lec 24 
Covariant derivatives - Lec 25
Application: Quantum mechanics on curved spaces - Lec 26 



Application: Kinematical and dynamical symmetries - Lec 28 

 A: I'll give some suggestions I've found useful as I've been recently studying some of these topics. These are just in the order they come to mind. I'll try link to amazon or somewhere else you can preview the books. Hopefully this is a start.
Firstly there's a range of highly referenced textbooks which are about Topology and Geometry for physicists:


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*C. Nash and S. Sen - Topology and Geometry for Physicists (Dover Books on Mathematics) - amazon. This is a fairly standard reference and tries to give a good feel and intuition for how mathematical ideas find applications in physics. Covers a bit of manifolds and algebraic topology like homotopy groups and homology and cohomology to fibre bundles to yang mills, morse theory etc. It is known to have the odd typo but it's even better to spot them as exercises :)(I just bought this edition the other day)

*M. Nakahara - Geometry, Topology and Physics, Second Edition (Graduate Student Series in Physics) amazon. This again is another standard reference I've come across. There's the odd typo which might be fixed in the second edition, but again covers a wide range of material - starts with a bit of QM, then onto topology and algebraic topology as above, manifolds and complex manifolds, some things in riemannian geometry, fibre bundles, gauge theories, characteristic classes etc. I took this out before with the intention to work through it, but I suggest skipping chapter $1$ and coming back at points later in the book unless you want to work through a lot of quantum mechanics and quantum field theory before he tells you what a map is in chapter $2$!

*T. Frankel - The Geometry of Physics: An Introduction amazon. As suggested by user: Bye_World, I actually have this book out of the library for more than the first time anyway. It actually covers a fair amount also, with some good examples and the odd in text exercise. It'll check most things on your list except possibly the representation theory and dynkin diagrams. Sometimes I wanted it to go into a bit more detail but it is a massive book, and so naturally it gives references where to look anyway!
As far as I am aware, those are the big three of books generally aimed at physicists. 
Another that might be worth mentioning is:


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*"Gravitation, Gauge Theories and Differential Geometry" by Eguchi, Gilkey, and Hanson, which is available here in pdf from Hanson's webpage. It's from Physics Reports but it's essentially a textbook, and while it jumps through topics a bit sharpishly, it's worth knowing of as it again is referenced a bit.


Before suggesting one or too maths text books there's also a couple of texts on representation theory:


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*Fuchs & Schwiegert - Symmetries, Lie Algebras and Representations: A
Graduate Course for Physicists (Cambridge Monographs on Mathematical
Physics) amazon. I've been meaning to read
this a bit more properly but it was recommended to me by a few
people. It does Rep theory with emphasis on symmetries, (as they come
up this way in physics), covering lie groups and algebras,
$\mathfrak{sl}(2)$, and cartan basis, dynkin diagrams etc.

*Carter, Segal, MacDonald - Lectures on Lie Groups and Lie Algebras (London Mathematical Society Student Texts) amazon. This was mentioned to me as a good resource by C. Nash for representations of Lie groups and algebras I believe. It's like $3$ smaller books in that 'Root systems and Lie algebras' are done by Carter, Segal does 'Lie groups' and MacDonald writes about 'Linear Algebraic Groups'. It'll lie groups and algebras and all that.
In terms of other books however for those topics, I found these useful:


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*Loring W. Tu - An Introduction to Manifolds (Universitext) amazon. This book is one of the most helpful I've found in solidifying the ideas needed for studying smooth manifolds (physicists like things to be $C^\infty$). It starts by introducing $C^\infty$ functions and preliminaries of the ideas applied to $\Bbb{R}^n$ before introducing manifolds in generality. It then works towards de Rham cohomology, covering a lot of ideas fairly thoroughly (I think). Very good in text and end of chapter exercises, and some hints at the back. It seems like a lot of things, exercises, theorems are mentioned precisely because they'll be used later on, so the book fits together quite neatly, fairly self contained with some appendices on the required linear algebra, topology, etc.

*Steenrod - The Topology of Fibre Bundles amazon. Although this is quite an old book, it really covers the basics of Fibre bundles well, and I enjoy reading it. It places a good emphasis on of course on topology, which isn't really emphasised so much in physics texts. Mostly the spaces are assumed to be nice enough anyway in physics, and sometimes the definition of a Fibre Bundle can be a bit lax or a bit too wordy. A good lot of proofs!

