Gracious Living


Compactness and the Heine-Borel Theorem
November 29, 2010, 20:57
Filed under: Math, Topology | Tags: , ,

Most of the topological work we’ve been doing has been in the area of constructing new topologies. We’re now ready to move on and look at their properties. We’re looking in particular for properties that are intrinsic to the topology: we want them to be preserved under homeomorphism (and arbitrary continuous maps, if possible) and not depend on other structures like a metric, a vector space structure, or a specific basis. Of course, it’s always nice to apply topological properties to such specific structures.

We’ve seen a couple examples, importantly connectedness and path-connectedness (and their local versions) and metrizability. Boundedness was a non-example — the interval (0,1) is homeomorphic to \mathbb{R} — and in fact we found a bounded metric for any given metric that generated the same topology. But the idea of a space being “small” is nevertheless compelling. In \mathbb{R}, for example, the Extreme Value Theorem states that continuous functions on a closed interval [a,b] attain both a minimum and a maximum, as opposed to approaching either asymptotically. This and other related theorems make doing calculus on closed intervals really nice. All these properties of closed intervals derive from a single topological one: compactness.

(Be warned: this is a pretty long post!  Mostly because I give a very careful proof of a theorem I consider important.)

Continue reading



Metrics
November 26, 2010, 12:14
Filed under: Analysis, Math, Topology | Tags: , , , ,

It looks like, in the homestretch, I’ve been unable to post every day, so I’m going to consider myself out of MaBloWriMo.  Which is a pity.  On the other hand, I’d prefer, in the end, to post better posts less frequently and on a wider variety of topics.

In the interests of total confusion, let’s discuss metrics, with have almost nothing to do with measures, despite the similarity in name!  Measures have to do with the sizes of sets of points, and are defined only on \sigma-algebras.  Metrics have to do with the distance between points, and are everywhere defined.  Metric topology was perhaps the earliest field of topology to be studied, and so it’s not surprising that a metric on a space will give you a natural topology.  Going the other way, from topologies to metrics, was a central problem of point-set topology in the 20th century.

Continue reading



Measures and Non-Measurable Sets
November 23, 2010, 14:22
Filed under: Analysis, Math | Tags: , , , ,

This is very late, but don’t worry, I’ll get another one up tonight.

One of the big lessons learned from the Banach-Tarski paradox is that even in something as simple as a unit ball, we can find sets of impossible or undefinable volume. In the discussion preceding the proof, I also mentioned the paradoxes surrounding length of fractal curves in \mathbb{R}^2 and area in \mathbb{R}^3. Together, these present us with a crisis: how can we characterize length, area, and volume? The answer to this crisis was developed around the turn of the century by heroes like Borel and Lebesgue, and it’s called measure theory.

Continue reading



Continuous Maps and Homeomorphisms
November 4, 2010, 16:53
Filed under: Math, Topology | Tags: , , , ,

If you’ve done any calculus, you’ve seen continuous functions.  If you haven’t, the concept isn’t that difficult to understand.  A continuous function is one that doesn’t jump around, instead moving smoothly from point to point.  Formally, we say a function f:\mathbb{R}\rightarrow\mathbb{R} is continuous at a point x if the limits of f(x_0) as x_0 approaches x from either side exist and are both equal to f(x).  If the term “limit” is unfamiliar, we can unpack the definition further: f is continuous at x if for every \epsilon>0, there is a \delta>0 such that whenever |x-x_0|<\delta, we have |f(x_0)-f(x)|<\epsilon.  This looks pretty nasty, but really all it’s saying is whenever x_0 is close to x, f(x_0) is close to f(x), and by moving x_0 closer, we can get the margin of error as small as we want.

Okay, and now we’ve hit that magic word “close” again, and you know what that means — we can extend this definition to functions between arbitrary topological spaces!

Continue reading



MaBloWriMo
November 2, 2010, 03:47
Filed under: Blog, Math | Tags: , , , , , , , ,

So it appears there’s this thing. It’s inspired by this other thing. As a wannabe NaNoWriMoer (I will have the time someday!), I do find the idea compelling: Siegel’s variation is to write a blog post of about 1000 words every day, on a single subject, ideally something the blogger knows little to nothing about.

So… I’ve been doing posts about every day, and they’ve been over a thousand words in general (going by the WordPress word count, which counts “words” in LaTeX formulas in some weird way). I don’t think this will be that difficult. On the other hand, I’ve been trying to teach math from the ground up, and since writing about a subject new to me will probably require more background than I’ve been assuming, I feel like it would be unfair to my readers, insofar as I hope to attract some.

The subjects I’d want to learn most of all in the context of a project like this are algebraic geometry (for which, small plug, Charles Siegel’s series looks excellent) or category theory (I’ve just gotten started on Mac Lane’s Categories). If I wanted to start from a high school background, however, my options are more limited: I feel like I could do some aspect of classical analysis, like Fourier analysis, maybe using Stein and Shakarchi’s Princeton Lectures vol. 1, which looks pretty neat. I’ve also just picked up a pretty simple-looking book on computability theory and logic from the co-op’s free pile, which isn’t my favorite subject, but what the hell. Finally, I got (from the same pile) a textbook on thermodynamics, which I know shamefully little about; it’s not really math, but I could give a more mathematical perspective, and it might be kind of fun.

The final option, which I’m leaning towards, is to continue laying the foundations of set theory and topology, but to do so every day. Maybe to work on conciseness and shoot at thousand-word rather than fifteen hundred-word posts. If anyone has any opinions or ideas, let me know!