Gracious Living


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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Group Actions
November 16, 2010, 15:00
Filed under: Algebra, Math | Tags: , , , , ,

Today we’re going to take the abstract group machinery we’ve been building up, and unleash it on some sets. When mathematicians say that “groups describe symmetry,” this is exactly what they’re talking about. Say we have a set, and some “symmetries” of that set. Pretty much any definition of “symmetry” will take it to be a bijection on the set, and we moreover expect to be able to undo symmetries, to compose them (associatively), and to use the identity map as a symmetry. These heuristics are just informal versions of the group axioms! The element we’ve been leaving out so far, though, is the set itself on which the symmetries are founded. We say that this symmetry group acts on this set. Below, let’s make this formal.

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Groups
November 11, 2010, 19:00
Filed under: Algebra, Math | Tags: , , , , ,

New topic!  To be mixed in with the topology.  We’ve been talking a lot about a strange thing called \mathbb{R} and some understanding of algebra will help us pin down what exactly this is.  (“Algebra” here is not the same as the “algebra” you learn in high school, by the way, though there is a relation.)  I’d also like to bring up group actions in the context of quotient spaces, and maybe start discussing the fundamental group of a topological space.

An introduction: Groups are one of the first things most math students learn in college, and probably one of the more  useful.  Though study of groups themselves is mostly viewed as a closed subject these days, the idea underlies many more current ones, and is the foundation of modern algebra.  The idea was essentially invented by the mathematician Évariste Galois, who used it to study the interaction between the roots and coefficients of polynomials1.  Shortly after, the idea of a continuous group of transformations (of, say, figures in geometry) was studied by people like Felix Klein and Sophus Lie; with discrete groups, problems were usually more combinatorial in nature, and study has historically centered on ways of efficiently categorizing and understanding large groups (such as “representing” them as linear transformations of a vector space, or writing them as constructions made of simpler groups, or as generators and relations, or so on).

The group concept is a highly abstract one, but once you begin to understand it, you see how wide its applications are.  Below the fold, let’s do some math!

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Product and Disjoint Union Topologies
November 8, 2010, 15:55
Filed under: Math, Topology | Tags: , , , ,

As the subspace topology is the “best way” to topologize a subset, the product and disjoint union topologies are the “natural ways” to topologize Cartesian products and disjoint unions of topological spaces.  Usually the disjoint union topology is only glossed over, while more time is spent on the product topology; I’m introducing them together in order to show you some of the similarities between them addressed by category theory.  The relationship is one of duality, something like the intersection and union of sets.  There’s this goofy mathematician way of putting “co” in front of dual constructions, so we could also call the disjoint union the “coproduct”.

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Subspaces, and more continuity
November 7, 2010, 06:19
Filed under: Math, Topology | Tags: , , ,

A subspace of a topological space X is a subset Y of X with the following topology: a set is open in Y if it is of the form U\cap Y, with U open in X.  We call X the ambient space, and this topology the subspace or induced topology.  It’s our first way to make old topologies into new ones.  Below the fold, I discuss its implications and a couple more factoids about continuous maps.

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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!

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Order and Topology
October 28, 2010, 18:41
Filed under: Math, Topology | Tags: , , , ,

Today I’m going to talk about two more or less unrelated ideas: a partial order on the set of topologies over a space, and topologies that “come automatically” with a totally ordered set. Hopefully my post will be a good deal shorter than last time.

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