# Real analysis with a non-standard topology

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

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.

• Hi Bryan, if I interpret your first paragraph correctly, analysis builds on some primitive topological set-up. My question is a little more specialized, suppose we start with a set say $\mathbb{R}$ and then construct two incomparable topologies on it. Do we have a feeling for what happens when we construct the rest of the required machinery on top of those two cases? In my example, do we get identical structures at completion with both the Sorgenfrey line and the standard topology? Will differences in the topology elicit a difference with metrics and linear structures?
– JEM
Jun 20, 2014 at 20:17
• It's not just built on top of it. It's more like an area that's built upon at least four different legs (topology, fields, linear algebra, and norms). And topology is blind to linear algebra and fields, and to a certain extent norms. The theory of Banach spaces pools these things together and tells them to work together in a systematic way (such as vector addition and scalar multiplication being continuous, and defining derivatives using the available norms and limiting processes).
– user123641
Jun 20, 2014 at 20:24
• You can construct incomparable topologies of course. The question is whether that new topology can be forced to work with field/vector/norm operations in the same convenient ways. For that kind of question, I'm not qualified to answer and you should seek out a differentiable topologist who really know their foundations.
– user123641
Jun 20, 2014 at 20:26
• Actually I do recall that any locally-compact, connected, topological field is either isomorphic to $\Bbb C$ or $\Bbb R$ (with their usual topologies and canonical operations). This severely limits the available analysis you can do over either of these fields.
– user123641
Jun 20, 2014 at 20:33
• On the other hand, there is a whole separate branch of $p$-adic analysis, with derivatives and power series and so on, based on the $p$-adic numbers. Very important to number theory. The $p$-adic numbers are not connected, in fact they are totally disconnected, but that's not a particularly important property for these purposes, apparently. Jun 21, 2014 at 3:36