Real analysis with a non-standard topology

Question

I have recently undertaken a self study of topology and am using Munkres Topology 2nd edition as the primary text. My background(theoretical chemistry & physics) is almost entirely void of any formal training in proof writing. I hope this struggle with Munkres will result in some marked improvements. I have some general questions regarding the concept of topology:

1. How does changing a topology affect other areas of analysis? For example, how would the study of calculus or real analysis change if I used the Sorgenfrey topology versus the standard topology on $\mathbb{R}$? Is my definition of derivative the same?
2. Is there a "correct" topology for studying physical phenomena? For example, how would modifying a topology affect numerical simulations of a dynamical system?
3. One of the topics in Munkres is a study of continuous functions on various topological spaces. Some functions are continuous in topology $\tau$ but not in $\tau'$. Does this help us "do" things like integrate over weird sets?
4. Is it possible to choose some non-standard topology for a symplectic space that would allow the integration of a Hamiltonian over some chaotic domain.

I suppose I am craving some feeling for what dividends will be paid for my time invested in general topology? Thx

Answer 1

There is a lot more to studying derivatives and calculus than simply topological spaces. Topological spaces give you basic set-up to study continuous functions, but the basic set-up of differentiable functions is encapsulated in Banach spaces which are topological structures with extra linear structure and ideas of distance. Also, to study integration 'correctly' you need the ground work of measure spaces. And to intertwine the integration with differentiation, your underlying space needs the structure of both a Banach space and a measure space.

If you're worried about whether you're going to get anything out of topology, don't worry-you will. For example, the Mean Value Theorem and Extreme Value Theorem, which are often initially presented as theorems of calculus or analysis, are actually theorems just from topological concepts (namely connectedness and compactness). The study of Point-set topology is a study in reverse-engineering. You'll be getting what properties of the real numbers (and other topological spaces) arise just from purely topological and continuous notions rather than differentiable, smooth, monotone, measurable, or linear notions.