Rethinking Topology (or a Personal Topologodicy)

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When I was originally introduced to topology, I simply accepted most of its properties as generalizations of ℝⁿ. I didn’t give it any serious thought until about a month ago when I read an excellent thread on math overflow about it. Since then, its been one of the things I often find myself thinking about when I’m trying to fall asleep. Given the amount of thought I’ve put into it, and the fact that I feel I should be answer questions like this about topology, given that it’s one of the areas of math I spend a lot of time on, I thought I’d write up my thoughts. They lent themselves well to being written in the form of an introduction to topology, so that’s what I did.

(After finishing this essay I decided to reread the MO thread. The first comment — not answer, a comment — mentions the Kuratowski closure axioms and closure axioms sounded like one might call what I came up with. Sure enough, they’re the exact same, down to the ordering. Are all attempts to make mathematical contribution’s this frustrating? I’m posting this because of the amount of work I put in, but there’s nothing new here.)

Consider 1 with respect to [0,1). It isn’t part of the set, but in a sort of intuitive sense it almost is. And knowing which points are `almost in’ a set gives us lots of information, for example notions of boundaries and connectedness.  Topology is based on us formalizing this notion of `almost in’ and once we formalize it, we can consider non-standard notions of being `almost in’ or apply these ideas to spaces that we don’t typically associate them with.

Separation Axiom Visualisations

A couple days a go, I saw some nice visualisations of separation axioms on Wikipedia. Unfortunately, it wasn’t a full set. Well, here is a full set (well, T0, T1, T2, T2 1/2, T3, T4, T5):

The Mandelbrot Set: Compact?

Several weeks ago, I read something on Wikipedia that shocked me: “The Mandelbrot set is a compact set.”

At first I didn’t believe it. How could the Mandelbrot set, in its infinite complexity, be compact?

Compactness Graph

Here’s the first revision of a graph of the implications of topological properties that I made:

(Click on it to see a better version!)

It’s mostly based off the stuff in Counterexamples In Topology (great book, BTW) but I did add some stuff (like Baire!) and merged/reorganised it. Diagram was made by Graphviz.

Most of the implications are trivial, but there are a few I haven’t prooved yet (most of the ones involving seperation axioms).

Limits and the Infinitesimal Number

I’ve been thinking about the infinitesimal number, , recently. In particular, that one could use it to evaluate limits.

What is a limit, really? I’ve been reading some topology recently and I think that it really is a function that returns an accumulation point (hint: these are alternatively known as limit points). More specifically, I believe that is an attempt to find a value such that is a limit point of the graph of the the function .

But I’ve digressed since the simpler, “It’s the value as we approach the point” is perhaps more useful to us…

Consider . How is this defferent from ? The difference is that we’re looking for the hypothetical value that the function is becoming (also the value of the point which any open set containing it intersected with the graph is not null), not its value when it is infinitly close. Consider : the difference is versus . So, we need to get rid of the infintesimal difference. Let represent the rounding of anumber to the nearest real number. Then,

Does this have any applications? I beleive it may provide a more elegant way to present Calculus.

Why can’t I take the first principles defenition of a derivative and:

? Because the defenition of a limit would give us zero, that being the closest real number. But instead, we could just say that a value is close to delta and use that. Not useing because I see no reason to use a random symbol when there is a logical one.

And we have a differential form, let’s add vectors (for more dimensions):

and for yet more clarity:

which is far less cumborsome than

And has almost identical properties (I suppose that since it is it will be forward facing, eg, instead of undefined).

Just some random thoughts.