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 is homeomorphic to — 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 , for example, the Extreme Value Theorem states that continuous functions on a closed interval 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.)
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.
Filed under: Algebra, Math | Tags: abelian groups, algebra, categorical, group theory, Math
Before looking at solvability and group classification, I want to mention a couple more ways of “building” groups. We’ve already seen how to find subgroups, and how to take the quotient by a normal subgroup, and how to find the direct product of a family of groups. Dual to the direct product is the free product, which generalizes the idea of a free group. The amalgamated free product is just a free product that we neutralize on the image of some map. Also, though the only really good example is the group of Euclidean isometries, the semidirect product is worth a more formal look. Finally, though it’s mostly terminology, I define the direct sum, which is useful for studying abelian groups.
Filed under: Analysis, Math, Topology | Tags: analysis, Math, metric topology, pretty pictures, topology
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 -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.
Filed under: Analysis, Math | Tags: analysis, banach-tarski, MaBloWriMo, Math, topology
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 and area in . 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.
Filed under: Algebra, Math | Tags: algebra, geometry, group theory, linear algebra, MaBloWriMo, Math
So I sort of left you hanging last time. We talked about equidecomposability, showed that was paradoxical under its own action on itself, and embedded into . From here, it just becomes a matter of putting all the steps together: first the sphere, then the ball minus its center, then the whole ball.
Filed under: Algebra, Math | Tags: algebra, geometry, group theory, linear algebra, MaBloWriMo, Math, topology
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 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.”
Filed under: Algebra, Math | Tags: algebra, categorical, group theory, MaBloWriMo, Math
Ugh, so, I’ve been really busy today and haven’t had the time to do a Banach-Tarski post. Since I really do want to see MaBloWriMo to the end, I’m going to take a break from the main exposition and quickly introduce something useful. There are a couple major ways of combining two groups into one. The most important one, called the direct product, is analogous to the product of topological spaces. I know this is sort of a wussy post — sorry.
Filed under: Algebra, Math | Tags: algebra, geometry, group theory, MaBloWriMo, Math, topology
Finite-dimensional vector spaces 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 with the following properties:
- Bilinearity: , where are scalars and is another vector, and the same for the second coordinate
- Positive-definiteness: , and it is only equal to when .
(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 . This is only if 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 . In particular, is right iff . In this case, we say and are orthogonal.
In Euclidean space, the inner product is the dot product: . This is primarily what we’re concerned with today, so we’ll return to abstract inner products another day.
Okay, it’s time to make a big leap forwards in terms of concreteness. The Banach-Tarski paradox makes a strong statement about that isn’t true about or . Now, we still don’t really know what is, but if we pretend we know what it is, we can say stuff about . Certainly, has the product topology of — but it has much more than this. It has an origin, for instance, and a distance function, and a way to measure angles. The distance function, in turn, allows us to define spheres and isometries (i. e. distance-preserving maps), which are both part of the statement of Banach-Tarski. All of these are summarized by saying that is a vector space.