Topology Questions: Isham's Modern Differential Geometry for Physicists

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In summary, the conversation discusses the book "Modern differential geometry for physicists" by Isham and its relevance to the study of general relativity. The concept of topological spaces, filters, filter bases, and open sets are also mentioned and defined. It is noted that one does not necessarily need to study differential geometry to prepare for general relativity, as other books cover the necessary material.
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I'm a physics student and I'm trying to work my way through Isham's Modern differential geometry for physicists. I guess the first question would be what you guys think of this book, does it cover all the necessary stuff (it's my preparation for general relativity)? Sadly I'm already having trouble with the introductory chapter on topology.. I'll post my questions in this topic.

Isham starts out by defining filters and neighbourhood structures on a set X. A topological space is then a special kind of neighbourhood space. The proof of the following theorem is left up to the reader, but this reader needs some pointers..

"A neighbourhood space (X,N) is a topological space if and only if each filter N(x) has a filter base consisting of open sets."

When a filter base consists of open sets, what consequences does that have for the filter it generates?
 
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You don't have to study differential geometry to prepare for general relativity. The standard books used for both undergrad and grad courses develop the geometry that is needed to understand the theory. It's surprisingly not that much. When you want to study a more advanced treatment of some of the topics in relativity, that book will come in handy.
 
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It would help us if you defined your terms. How are topological spaces defined? Filters? Filter bases? Open sets? etc. etc.
 
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Ok, here are all the definitions that have been used:

The power set P(X) of a set X is a lattice, with [tex]A \preceq B[/tex] defined as [tex]A \subset B[/tex]. Now a subset U of P(X) is an upper set if [tex]\forall a,b \in P(x) [/tex] with [tex]a \subset b[/tex], [tex]a \in U [/tex] implies [tex]b \in U [/tex].

A filter F on X is a family of subsets of X such that F is algebraically closed under finite intersections, F is an upper family, and the empty set is not in F.

A filter base D is a family of non-empty subsets of X such that if [tex]A,B \in D[/tex], then there exists [tex]C \in D[/tex] such that [tex]C \subset (A \cap B)[/tex]

A neighbourhood structure N on X is an assignment of a filter N(x) to each [tex]x \in X[/tex], all of whose elements contain the point x. The pair (N,X) is then called a neighbourhood space.

Given a neighbourhood space (N,X) and any set [tex]A \in X[/tex], a point x in X is a boundary point of A if every neighbourhood of x intersects both A and the complement of A (a neighbourhood of x is an element of N(x)). An open set is then a set that contains none of its boundary points.

I suppose "each filter N(x) has a filter base" means that the filters N(x) are generated by those bases. In that case, here's the final definition:
If D is a filter base, then [tex]\uparrow(D) := \{B \subset X| \exists A \in D \ such \ that \ A \subset B\} [/tex] is a filter, which is said to be generated by D.

Edit: forgot one.. A topological space is a neighbourhood space (X,N) in which, for all x in X and for all M in N(x), there exists [tex]M_1 \in N(x)[/tex] such that, for all [tex]y \in M_1[/tex], [tex]M \in N(y)[/tex].
 
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1. What is the main focus of Isham's Modern Differential Geometry for Physicists?

The main focus of Isham's Modern Differential Geometry for Physicists is to provide a comprehensive introduction to differential geometry, with a specific emphasis on its applications to theoretical physics.

2. Is prior knowledge of physics required to understand this book?

Yes, a basic understanding of physics, particularly classical mechanics and electromagnetism, is necessary to fully grasp the concepts presented in Isham's Modern Differential Geometry for Physicists.

3. What level of mathematics is needed to understand this book?

A strong background in advanced mathematics, including calculus, linear algebra, and differential equations, is essential to fully understand the concepts presented in this book.

4. How is this book different from other textbooks on differential geometry?

Isham's Modern Differential Geometry for Physicists is unique in its focus on the applications of differential geometry to physics. It also includes a thorough introduction to the mathematical formalism of differential forms, which is often used in modern physics theories.

5. Is this book suitable for self-study?

While this book can be used for self-study, it is recommended to have a background in physics and advanced mathematics, as well as access to a tutor or instructor for clarification and guidance.

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