Gracious Living

November 3, 2010, 04:58
Filed under: Math, Topology | Tags: , ,

A quick post on topology because I’m stuck on my homework.

A neighborhood of a point is an open set containing it. This looks like a pointless definition, but it’s used in a very useful way: whenever I say “neighborhood,” I’m imagining the open set to be small.  I’m really thinking of a set of “close” points, like in the first post on topology.

Here’s an easy application: if a set of points includes a neighborhood of every point, then it is equal to the union of these neighborhoods, which are all open, so it must be open.  Conversely, any open set includes a neighborhood of each of its points: namely, itself.  Thus, a set is open if and only if it includes a neighborhood of each of its points. Taking the set itself as a neighborhood of each point is sort of cheating, but for “nice enough” spaces (we will make this formal), we’ll always have small neighborhoods of each point.

In essence, what this means is that points in open sets have some amount of “wiggle room.”  Another way to say this is that the condition of being in a given open set is stable: it is preserved under small perturbations.  Of course, “wiggle room” might not always look the way you think it does: in the lower limit topology, it’s entirely possible that a point can only be “wiggled” towards the right.

One theme of topology is the interaction between “local” and “global.”  Global properties are those that describe the entirety of a space; some pretty intuitive examples are “bounded” (which we’ll later formalize to “compact”) or “connected.”  Local properties are roughly those that can be said about a neighborhood of every point.  The manifold condition is a pretty simple example (again, we’ll make this formal): a sphere is certainly not a plane, but on a local level, when you “zoom in” to a single point, it looks very similar to one.

Another example is the local version of a basis:  A fundamental system or basis of neighborhoods at a point x\in X is thus a set \mathcal{U}_x of neighborhoods of x such that every neighborhood of x includes an element of \mathcal{U}_x.  A basis gives us a basis of neighborhoods at every point just by taking the subset of basis elements that contain that point; likewise, if every point has a basis of neighborhoods, then their union is still a basis.  Sometimes, we can get smaller bases of neighborhoods than those given by the global basis: in \mathbb{R}^n, the set of open balls with centers at a point x is a basis of neighborhoods at x, for example.

As I’ve remarked on, a second theme of topology is the ability to express analytical concepts without measurement.  For example, given a set A in a topological space, we define a limit point of A to be a point x such that every neighborhood of x contains a point of A (which is not x, if x\in A).  Recall the definition of the limit of a sequence in \mathbb{R}: x is the limit of (x_1,x_2,\dotsc) if for any \epsilon>0, we can choose a positive integer N such that |x_n-x|<\epsilon for n>N.  Prove that the limit of a sequence in \mathbb{R} is a limit point of the sequence in the Euclidean topology; by extending the expression |x_n-x| to denote distance in \mathbb{R}^n, prove the same is true in \mathbb{R}^n.  If you can, show that this is the only limit point of the sequence.  Which sequences have limit points in the lower limit topology?

Limit points also generalize the idea of boundary.  Briefly, the interior of a set A, written int(A), is the union of all open subsets of A, and the closure of A, written \overline{A}, is the intersection of all closed sets including A.  The boundary of A, written \partial A, is the closure minus the interior (I’m convinced the “del” symbol is used here because it makes Stokes’ Theorem look epic).  Hopefully, these words agree with your intuition — as usual, it’s good to check them on the alternate topologies we’ve seen (cofinite, lower-limit, discrete, indiscrete…).  The following statements hold, and you should prove as many of them as you can:

  • The interior of a set is the set of points which have neighborhoods in that set.
  • The closure of a set is the union of that set and all its limit points.
  • The interior of an open set is itself, and the closure of a closed set is itself.
  • Thus, a set is closed if and only if it contains all its limit points.
  • The boundary of a set is then the set of limit points not in the interior; alternately, it is the set of points that are limit points for both the set and its complement.

The early topologists treated these ideas more algebraically than analytically, viewing closure, interior, and so forth as operators acting on a set; I plan to give an axiomatic development of topology along these lines soon.  Here’s a super-cool problem posed by Kuratowski in the 1920’s that I saw on a problem set freshman year:

Problem (Kuratowski’s 14-set theorem).  Prove that the number of distinct sets you can get by starting with a set in a topological space and applying only the operations of “closure” and “complement” is 14.  Find a subset of Euclidean space \mathbb{R} that gives you 14 distinct sets.

(Hint for the first part: applying closure twice in a row is the same as applying it once, and applying complement twice in a row is the same as not applying it at all.  You’ll have to alternate them, and at a certain point a cancellation will occur.  Hint for the second part: the answer will be a union of a bunch of disjoint sets, one of which will involve something like \mathbb{Q}.  When you’re done, here’s an interesting extension; it defines some new terms, but I don’t think it’s too hard for anyone who’s gotten this far.)

Oh, and now that you know about countability, we can define some new topologies.  I like building up a useful array of examples for the future, and I hope that you’ll be able to check your topological intuition about an idea by experimenting with these examples.  Like the cofinite topology, we define the cocountable topology by declaring a set to be open if and only if it has countable complement or is \emptyset.  The underlying set can be anything, but if it’s countable, you get the discrete topology; \mathbb{R} is a pretty good choice.  The K-topology is the topology on \mathbb{R} generated by the union of the usual basis of intervals (a,b) and the collection of sets (a,b)-K, where K=\left\{1,\frac{1}{2},\frac{1}{3},\dotsc\right\}.  Show that in the cofinite topology on \mathbb{R}, every point is a limit point of every sequence; in the cocountable topology, no sequence has a limit point; and in the K-topology, a sequence has a limit point if and only if it is not a subsequence of K.

I have about two set theory posts to do (choice and ordinals), and then a bunch of topology.  I’ll take a break from the usual topology developments to talk more about closure as an operator and the Kuratowski approach to topology.  Comment if you like or dislike what you see!

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