Gracious Living

The Construction of the Reals: Metric Completion and Dedekind Cuts
March 6, 2011, 21:31
Filed under: Algebra, Math, Set Theory, Topology | Tags: , , , ,

It looks like I’m getting views now, which is surprising.  I’ve been pretty busy with schoolwork, but I really want to get this blog up to speed, particularly because I’d like to start discussing things as I’m learning about them.  I’d also like to make more non-mathematical posts, but maybe these are best left to a separate blog?  Thoughts?

Our first example of a field was the field of rationals, $\mathbb{Q}$.  Recall that this was the field of fractions of the integers, which were in turn the free abelian group on one generator with their natural multiplication.  But now it appears that we’re stuck.  While we intuitively know what $\mathbb{R}$ should be — it’s a line, for crying out loud — there seems to be no algebraic way of “deriving” it from $\mathbb{Q}$.  A first guess might be to add in solutions of polynomials, like $\sqrt{2}$ as the solution of $f(x)=x^2-2$, but not only does this include some complex numbers, it also misses some real numbers like $e$ and $\pi$.  (We call such numbers — those that aren’t solutions of polynomials with rational coefficients — transcendental.  It’s actually quite difficult to prove that transcendental numbers even exist.)

Instead, we turn to topology.  Below, I give two ways of canonically defining $\mathbb{R}$, one using the metric properties of $\mathbb{Q}$, one using its order properties.  I found this really interesting when I first saw it, but I can’t see it interesting everyone, so be warned if you’re not a fan of set theory or canonical constructions.  One of the topological techniques we’ll see will be useful later, but at that point it’ll be treated in its own right.

Countability Axioms
December 31, 2010, 01:55
Filed under: Math, Topology | Tags: , , , ,

So far we’ve seen two basic families of properties of topological spaces.  The connectedness axioms tell us in what ways it is possible to break our space apart into pieces.  The compactness axioms tell us how bounded the space is.  What we’re going to look at today is a set of axioms that deal with cardinality.  It should be mentioned that topology, for the most part, doesn’t really care about large cardinals — at most, we’re dealing with $\aleph_1$, the cardinality of our favorite counterexample $\omega_1$, and $c$, the cardinality of the reals.  These are equal if we accept the continuum hypothesis, and in either case we often talk about them in terms of countable subsets — sequences and the like.  The reason that countability is so important is that the properties we’re about to study are typical of metric spaces, and metrizability is a central question of point-set topology.

Limit Point and Sequential Compactness
December 30, 2010, 02:51
Filed under: Math, Topology | Tags: , , ,

Back to topology.  The interesting thing about compactness, as I see it, is that its definition isn’t very intuitive.  We want to talk about what are basically “closed and bounded” sets without really using closedness, which doesn’t behave well with subspaces, or boundedness, which doesn’t behave well with anything.  At the time this idea came about in the early part of this century, the mathematicians ssewho invented it (Borel, Weierstrass, and Bolzano, mainly, I think) messed around with a couple other definitions but they turned out to rely on intuition from Euclidean space, and so not be nearly as useful.  In this post, we’ll study those other definitions.

I debated with myself for a long time about what to do first: this, the separation axioms (which measure how good the topology is at distinguishing disjoint sets), or the countability axioms (which measure, well, how countable the topology is).  In a sense, each one depends on the other two for examples and theorems, and in this case, though it’s easy to show that all the properties I’m listing are different, we’ll need to refer to the other properties to discover when they are the same.  So I expect this post to be brief; after we have all the other properties defined, it’ll be easier to talk about what implies what.

Differential Geometry and the Sphere Theorem
December 16, 2010, 20:38
Filed under: Topology, Uncategorized | Tags: , , , ,

I just wrote a paper about the Sphere Theorem for my differential geometry class.  Since I can’t really get it out of my head, I thought it’d be fun to use to give a brief overview of differential/Riemannian geometry, in more or less layman’s terms.  This is somewhat of a departure from my normal posting style: though you’ll get more out of this if you’ve understood the topology posts so far, I’ll try to write in broad enough strokes that calculus is the only prerequisite.  Really, the most complicated idea behind this is the idea of a map from $\mathbb{R}^m$ to $\mathbb{R}^n$ being differentiable.  I also refer to vectors, but I think that that’s a pretty intuitive concept, in general.

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.)

Quotients of Topological Spaces
November 28, 2010, 20:06
Filed under: Math, Topology | Tags: , ,

The last way to induce a topological space is by taking a quotient.  This can be compared to taking a quotient of groups in the same way that the product topology corresponds to a direct product of groups, the subspace topology corresponds to a subgroup, and so on.  The quotient topology is a bit less intuitive than the other constructions we’ve done, but once you get the hang of it, it turns out to be very useful.

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.