Gracious Living


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.

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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.

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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.

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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.)

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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.

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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.

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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.

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Banach-Tarski part 1
November 21, 2010, 06:33
Filed under: Algebra, Math | Tags: , , , , , ,

Okay, here’s the moment you’ve been waiting for: the proof of the Banach-Tarski Paradox.  Here’s what the paradox says:

Theorem (Banach-Tarski).  There are a finite number of disjoint subsets of \mathbb{R}^3 whose union is the unit ball, and such that we can apply an isometry to each of them and wind up with disjoint sets whose union is a pair of unit balls.

Or “we can cut a unit ball up into a finite number of pieces, rearrange them, and put them back together to make two balls.”

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Isometries of Euclidean Space
November 18, 2010, 21:06
Filed under: Algebra, Math | Tags: , , , , ,

Finite-dimensional vector spaces \mathbb{R}^n come packed with something extra: an inner product.  An inner product is a map that multiplies two vectors and gives you a scalar.  It’s usually written with a dot, or with angle brackets.  For real vector spaces, we define it to be a map V\times V\rightarrow\mathbb{R} with the following properties:

  • Symmetry: \langle x,y\rangle=\langle y,x\rangle
  • Bilinearity: \langle ax+bx^\prime,y\rangle=a\langle x,y\rangle+b\langle x^\prime,y\rangle, where a,b are scalars and x^\prime is another vector, and the same for the second coordinate
  • Positive-definiteness: \langle x,x\rangle\ge 0, and it is only equal to 0 when x=0.

(I’m going to stop using boldface for vectors, since it’s usually clear what’s a vector and what’s not.)  One of the uses of an inner product is to define the length of a vector: just set \|x\|=\sqrt{\langle x,x\rangle}.  This is only 0 if x is, and otherwise it’s always real and positive because the inner product is positive definite.  Another use is to define the angle between two nonzero vectors: set \langle \cos\theta=\frac{\langle x,y\rangle}{\|x\|\|y\|}.  In particular, \langle \theta is right iff \langle x,y\rangle=0.  In this case, we say x and y are orthogonal.

In Euclidean space, the inner product is the dot product: \langle (x_1,x_2,\dotsc,x_n),(y_1,y_2,\dotsc,y_n)=x_1y_1+x_2y_2+\dotsb+x_ny_n.  This is primarily what we’re concerned with today, so we’ll return to abstract inner products another day.

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Local Connectedness and Some Weird Counterexamples
November 10, 2010, 14:25
Filed under: Math, Topology | Tags: , , , , ,

Last time we talked about connectedness and path-connectedness and showed that though all path-connected spaces are also connected, connected spaces are not in general path-connected. There is a partial remedy to this problem, found in the local versions of the connectedness properties.  Recall that a local property says something about neighborhoods of each point, rather than the space itself or its open sets.  Below the fold, I talk about local connectedness and local path-connectedness, and bring up a couple examples and counterexamples, including one of my favorites, the Conway base-13 function.

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