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