﻿This is my first attempt at explaining mathematical physics.


The goal is to understand spacetime. We know spacetime is a 4d topological manifold with extra structure.


The first step is clearly to understand topology.


It’s also probably a good idea to study a little bit of category theory to understand the idea of structure-preserving maps, which comes up again and again in mathematics, for example with vector spaces, groups, topological spaces, and others.


But to study category theory, you first need at least a few examples.


So let's start with topology.


Why do we care about topology? Topology is needed to understand continuity, assuming the least possible structure.


The reason we need continuity in physics is quite clear: we want our particles to move continuously from one place to another: which is intuitively to say that we don’t want our particles in space to suddenly “jump” from one place to another.


We use topology to enforce this condition.


Now what do I mean when I say topology is the framework with the “least possible structure” to talk about continuity? Well basically the point is that there are alternative ways to talk about continuity, but that they’re less general then topology. An example goes a long way.


Remember in calculus class, how we defined the continuity of a curve. Well basically the structure assumed in that case is that of a standard euclidean metric space: you can tell the distance between two points. In this case we can clearly talk about continuity, given the fact we assume that:
* The space were in is R^n
* We have a metric d:PxP->R which tells us the distance between two points
What I mean is that topology requires the minimal structure while allowing us to still talk about continuity; basically that we don’t need to assume the previous two things.


We still will be making a similar choice though. Except the choice must be made clear right at the start. The two choices above will end up accompanying similar informations:
* The choice of set X
* The choice of topology on that set O
In fact, because topology is the most general framework for talking about continuity, it also encapsulates the definition above.


One objection readers might have is: why do we need a more general case from the standard metric on R^n approach. And the answer is that eventually spacetime will turn out not to be R^n, so it makes no clear way to define a metric (at least not immediately, later on we do indeed define a metric implicitly from the choice of connection). So this justified the more general study of continuity on any object: topology.


So topology can essentially be thought of as a protocol for talking about the continuity of any space X given a choice of topology on that O. And you’re damn right correct if you suppose that the O will contain all the information about the choice of what is close to what on X.


That last statement makes sense only to me, so let me reexplain with an example: consider X is {1, 2, 3}, the choice of if 1 is close to 2, or if 2 is close to 3, or if everything is close to everything, or if nothing is close to nothing, is all selected in the choice of O. If, for example, you choose O to be the chaotic topology, then nothing is close to nothing. If you choose O to be the discrete topology, then everything is close to everything. But notice that when everything is close to everything, or nothing is close to nothing, you’ve essentially added no useful information.


Anyways the actual choice of O is restricted to some degree, essentially for the purpose of avoiding paradoxical situations. Let me explain the definition of topology in terms of paradox avoidance.


Some language: open set means an element of O.


0) O is a subset of the powerset of X. This can be thought of as the type selection of O: what kind of data can O be so that it makes sense. For example if you ask “how many provinces in canada”, you better associate a type “positive integer”, because answering something else doesn’t make sense (like “false”). So the idea is that for O you’re essentially selecting which subsets of X are “close”. If points p and q are close, then you will put {p,q} in O.


1) empty set & X are in O. This is done to avoid inconsistencies with 2) and 3)
2) finite intersections of open sets are in O. This can be thought of as “If A is close to B, and B is close to C, then A must be close to C”
3) arbitrary unions of open sets are in O. This can be thought of as “If A is close to B, and B is close to C, then ABC must all be close to each other”.


The general idea of “continuity” in topology is that the more times two points have open sets in common, the “closer” they are. This explanation breaks down when inspected closely, but is generally a good way to see the link between the protocol that is topology, and continuity, the thing it tries to define extremely generally.


Tip: a lot of times when trying to understand something in topology, it makes sense to look at the case of R^n equipped with the standard topology. This is because topology was essentially created to satisfy all R^n standard topology stuff.


For example, consider I said “finite intersections” for 2). Why is that? Looking at the case of R^n, we see it’s because if you are allowed to take infinitely many intersections, you can eventually arrive at a point, which is not good because the whole point of standard topology is that only venn diagrams with soft edges can be open sets, so standard topology would break if 2) was allowed to be arbitrary intersections.


Ok so what’s next: well we need to actually talk about what it means for a curve to be continuous.


Tip: whenever I say curve, I think of f: R->M, and whenever I say function, I think of f: M->R, and whenever I say map, I think of f: M->N.


This is because a curve associates a parametric variable to every point of the space M. A function associates a temperature on every point of M. And a map takes every point of M to a point in N.


This post was partially written on december 29th


So what does it mean for a curve to be continuous?