Chapter 24 of 24 - Fractals

Finally reached the end of the materials.

First section introduced some examples of fractals. Second section was "Using the Contraction Mapping Theorem" which introduced the notion of distances between compact sets, ultimately the "Hausdorff Distance", and the notion of contraction mappings that kind of translate your base shape down to the next iteration of the fractal. Last section was on dimensions of fractals, which enforced a lot of logarithm revision.

Again, this section was merciful when compared to the average difficulty of some of these chapters.

Suddenly I'm putting the finishing touches on my last ever written assignment. Nearing the summit of Maths Mountain.

From here on out it's mostly revision. Going over past assignments and preparing for my last ever maths exam. Let's see what useful little bits of information I can find and document over the next month or so...

Chapter 23 of 24 - Completeness

Loving the irony that the penultimate chapter is on "completeness".

First two sections covered completeness of metrics as defined by sequences, including examples of basic well-known metrics that are complete.

Third section was on the contraction mapping theorem, which seemed fairly profound. It allows you to show that generalised differentials or integrals have unique solutions. I needed to speed through this section, and I wonder whether a review of past papers will reveal that I need to spend more time on this. Like a lot of this, as some of the answers are so lengthy, I do wonder how feasible it is that any of it will be in the exam.

The last section covered methods by which you can find a completion of a metric. This presented a framework that was really interesting, but offered no examples. There was even an admission that the process they'd presented was lengthy and with examples and not mention of it in the assignment, I can only guess it too is not covered in the exam.

So. One quick review of the ongoing assignment and I'll finally be on to my last chapter. Ever.

Chapter 22 of 24 - Compactness

The first section here was a stern-sounding "Two Important Theorems". It covered the boundedness theorem and the extreme value theorem, both I've covered before in previous pure maths modules, but this related them directly to metric spaces.

Second section decided to introduce compactness through the concept of sequential compactness, which essentially describes the "d-closed and bounded"-ness of a set. Though is inevitably difficult to learn quickly because of strange concepts involving sequences in metric spaces.

Third was "functions and sequential compactness", which was even more strange as it extended section two to also cover all functions in C[0,1], and various hard-to-understand \epsilon-\delta theorems.

Section four was "compact metric spaces" and the notion of covers and sub covers. Far easier to understand, but depending on the set it can be tricky (for example) to find a finite sub cover from an infinite cover. With a bit of practise you get the trick though.

Section five was how sequential compactness and compactness are equivalent for metric spaces, and the last section covered unions and intersections of compact spaces. Again, these sections I sped through as section 5 was largely theorems, corollaries and lemmas, and section 6 was largely unexamined.

Chapter 21 of 24 - Connectedness

First section here was on homeomorphisms, which is essentially a brief intro in the foundations of topology. I've been wanting to study topology for years, but as is the trend, I didn't have time to dwell. Again, we don't get examined on topology, so this will be something I'd want to come back to after my final exam.

Next up was "closed and open sets revisited", which is practically exactly what you expect.

The third section introduced the concept to connectedness properly, and the fourth went on to explore connectedness in Euclidean space.

At this point, I'd learned enough to tackle the assignment questions. I didn't read through the last two chapters, "path-connected spaces" and "the topologists cosine", but I didn't feel too guilty about this. I'm already vaguely familiar with path-connected spaces, and the last section seemed to be unlikely to be examined. I'll review both of these sections during my revision period though.

All in all, this chapter was mercifully easy to understand.