I would like to learn differential geometry the mathematicians way, not the physicists way. I usually just gloss over passages about Hausdorff spaces, second countable, paracompact and things of that nature and I would like to stop doing that.Demystifier said:A physicist, even a mathematical physicist, can live without ever reading Munkres or anything equivalent. Books entitled "Topology for physicists" or something alike should be sufficient, and such books do not pay much attention to things studied in Munkres-like books. But of course, if you want to know more topology than a physicist really needs to know, you are welcome to read Munkres or anything else which you find interesting.
That's fine, but I don't think that learning math the non-physicist way can be essential for physics. Maybe useful, complementary, or deepening, but not essential.cpsinkule said:I would like to learn differential geometry the mathematicians way, not the physicists way.
The purpose of studying Munkres Topology is to gain a solid understanding of the fundamental concepts and principles of topology. This branch of mathematics is concerned with the study of properties that are preserved through continuous deformations, such as stretching or bending, and has applications in many areas of science and engineering.
The essential chapters of Munkres Topology include Chapter 1: Set Theory and Logic, Chapter 2: Topological Spaces and Continuous Functions, Chapter 3: Connectedness and Compactness, Chapter 4: Countability and Separation Axioms, and Chapter 5: The Tychonoff Theorem.
Chapter 1 of Munkres Topology covers set theory and logic, including topics such as sets, functions, relations, and cardinality. These concepts are fundamental to understanding topology and provide a solid foundation for the rest of the book.
Studying connectedness and compactness in topology is important because these concepts play a crucial role in understanding the structure of topological spaces. Connectedness refers to the property of a space being in one piece, while compactness refers to the property of a space being finite in some sense. These concepts have important applications in analysis, geometry, and other areas of mathematics.
The Tychonoff Theorem, also known as the Tychonoff Product Theorem, states that the Cartesian product of any collection of compact spaces is compact in the product topology. This theorem is significant because it provides a powerful tool for constructing new compact spaces from existing ones, and has important applications in functional analysis and topology.