Gracious Living

October 28, 2010, 09:49
Filed under: Math, Topology | Tags: , , ,

What we’re going to do today is a degree more abstract that what we’ve seen before.  Unlike sets, which everybody really thinks of as piles, functions, which everybody thinks of as algorithms, and equivalence relations, which we proved you could think of as partitions, topologies don’t have an easy built-in metaphor.  Higher math commonly defines these abstract lumps that end up having nice properties, but when you learn about them, it’s difficult to get past the fact that under the properties, they’re only abstract lumps.  I’d say topological spaces are the first abstract lump most students encounter — yet by being abstract, they create a foundation for dozens of more specific fields.  There is a metaphor you can use to understand them, though as we’ll see, you don’t always want to use it, and either way it takes some thinking about.

various bases about a point

All three shapes give the same idea of closeness.

The metaphor underlying topology is closeness.  Take three points in \mathbb{R}^3 — that is, standard Euclidean 3-space, with x-, y-, and z– axes and all the rest.  How do you decide which of the second and third points is closest to the first?  Clearly, you just measure their distances.  What if I just asked you to tell me which points in space were close to the first one?  You’d probably say something in terms of distance as well: “the points inside this ball are all within 0.001 units of distance from our point, so they’re pretty damn close.”  Maybe you would decide to formalize it, calling the points inside a sphere of radius rr-close” to the center, and noting that every r-close point is also s-close for s>r, so that we can choose which of two points is closer by seeing which one satisfies more closeness conditions.

“But this is just the same as finding distances!”  Sure it is.  But check this out — we don’t need to use spheres.  We can instead r-closeness to mean “contained in a cube of side length r centered around our point.”  Or we could use irregular tetrahedrons.  Now the distance function and closeness sets are utterly separated, but if one point is closer than another in any of these definitions, it is closer in all of them!

Ant on a Torus

How does the ant know what kind of surface it's on?

The basic idea of topology is that we can define this kind of “closeness” without even needing to measure distance at all.  Believe it or not, problems with distance-measuring can crop up in even the most familiar of spaces — our own.   Though distances are easy to measure in “flat” Euclidean space, our space is actually curved by mass and other relativistic forces.  So finding the distance between two points theoretically requires us to find a shortest path between them.  But we can study the curvature of space and find its “shape”, using topology, without having to measure distance at all.  If this is difficult to visualize in 3 dimensions, imagine an ant on a surface.  If the surface is large enough, the ant can’t really see what it looks like — but we’ll show how the tools of topology can allow him to figure it out from the inside.

The math we’ll be using is all founded in set theory.  Consider a set X, whose elements we’ll call “points.”  A topology on X is a set \mathcal{O} of subsets of X (that is, a subset of \mathcal{P}(X) satisfying the following conditions:

  • \mathcal{O} contains X and \emptyset1.
  • For any finite subset \{U_1,\dotsc,U_n\}\subset\mathcal{O}, we have U_1\cap\dotsb\cap U_n\in\mathcal{O}.
  • For any subset, finite or infinite, \{U_\alpha\}\subset\mathcal{O}, we have \bigcup_\alpha U_\alpha\in\mathcal{O}.

When we equip X with a topology, we call the result a topological space.  Sometimes we just call the set X a topological space, when the topology is known.We call elements of \mathcal{O} open sets and their complements closed.  Watch out: a set can be neither open nor closed, and it can be both (clopen).  Insert a joke here about a topologist who is unable to deal with a door.

The big question, though, is: what the hell does this actually mean?

The answer, or at least its beginning, is that open sets give you the same “closeness” information we talked about up top.  Let’s again look at Euclidean space for motivation, this time \mathbb{R}^2 because I want to draw a picture.  You’ve probably seen “open and closed sets” in \mathbb{R}^2 before, probably first when studying inequalities.  Informally, a closed set contains its boundary and an open one does not.  I’ve drawn open sets with dotted lines to represent this.Examples of union and intersection in the Euclidean topology

As you can see, two closed sets can intersect each other in anything as small as a point.  But two open sets can only intersect in another open set.  If you take the union of a bunch of open sets, no matter the bunch, it still won’t contain anything close to a boundary — you can incrementally approach one but never contain it.  On the other hand, the intersection of an infinite bunch of open sets can be a point — just let U_n be the open ball of radius 1/n about some point.  And we really don’t want this to be open.  If points are open, then every set, being the union of points, is open, which is icky and boring.

In this case, it is more useful to think about the boundary than the open set axioms.  In fact, there are many other equivalent axioms for a topology.  I hope to look at at least one of them when I talk about boundary.  Though the open set axioms are less intuitive, we use them because they’re easier to do proofs and express topological concepts with.  One of the amazing things we’re going to discover is that familiar notions from analysis, like limits and continuity, are naturally expressible in purely topological terms, without the need for measurement.

For now, two ways to simplify the open set axioms.  It is very rare that an author outright defines a topology \mathcal{O} by listing its elements.  If the topology is not induced from other structures in ways we’ll talk about later, it is common to talk about it in terms of a basis.  A basis is a set \mathcal{B} of subsets of X such that:

  • \bigcup\mathcal{B}=X, and
  • the intersection of any finite number of elements of \mathcal{B} is in \mathcal{B}.

Then we can form a set \mathcal{O}_{\mathcal{B}} whose elements are all of the unions (finite and infinite) of elements of \mathcal{B}.  You should check that \mathcal{O}_{\mathcal{B}} is a topology; it is called the topology generated by the basis \mathcal{B}.  How to go the other way?  Given a topology \mathcal{O}, we define \mathcal{B}_{\mathcal{O}} to be any subset of \mathcal{O} such that, given x\in X,U\in\mathcal{O}, there is an element of \mathcal{B}_{\mathcal{O}} that contains x and is a subset of U.  Prove that \mathcal{B}_{\mathcal{O}} is a basis and generates the topology \mathcal{O}.

We could go one step further and define a subbasis for \mathcal{O}  to be any subset of \mathcal{O} such that every element of \mathcal{O} is a union of finite intersections of its elements (equivalently, such that a basis for \mathcal{O} is composed of finite intersections of its elements).  This will be “smaller” than a basis but generally more difficult to state.

For your perusal today, I give you three lines, one nice, one medium, and one naughty.  The nice line is just \mathbb{R} with the Euclidean topology.  We’ve talked so much about the Euclidean topology that it’s time to define it.  The Euclidean topology on \mathbb{R}^n is that generated by the basis of all open balls centered around every point.  (By the way, I’m going to call the ball of radius r around x by the notation B(x,r)=\{y:|y-x|<r\}.)  On \mathbb{R}, balls are just open intervals, so open sets are unions of open intervals.  If we consider the rays (-\infty, a) and (\infty, b) to be called intervals, then we can instead say that “open sets are unions of disjoint open intervals.”  A subbasis for this topology is given by \{(-\infty,a),(\infty,b):a,b\in\mathbb{R}\}: do you see why?

A quick digression to higher dimensions: in \mathbb{R}^n, we can define a cube to be the Cartesian product of open intervals: (a_1,b_1)\times\dotsb\times (a_n,b_n).  As a non-easy exercise, prove that the topology generated by these guys is equal to the Euclidean topology generated by the balls.  (Hint: start by showing that any ball around a point contains a cube around that point, and vice versa.)  This allows us to give a subbasis for \mathbb{R}^n as the set of “half-spaces” \mathbb{R}\times\dotsb\times\mathbb{R}\times (a_i,\infty)\text{ or }(-\infty, b_i)\times\mathbb{R}\times\dotsb\times\mathbb{R}.

Here is the medium-weird topology for \mathbb{R}.  The closed sets here are the finite sets, so the open ones are those with finite complement.  (What would happen if it was the other way around?)  We call this the cofinite topology, where “cofinite” means “with finite complement.”  Prove that this is a topology.  When are points “close”?

The weirdest topology is the Sorgenfrey line, also known as the half-open topology.  This is generated by the basis whose elements are intervals of the form [a,b).  Show that open intervals (a,b) are open, and then show that intervals [a,b) are closed as well as open.  What about intervals (a,b]?  Can you come up with a subbasis for this topology?

Next time, I will talk about either cardinality, Kuratowski closure, continuity, or ordering. Be prepared for a surprise!

1This condition is technically redundant. No matter what sets we know are in \mathcal{O}, we know for sure that the union or intersection of 0 of them is in \mathcal{O}, by applying the bottom two conditions with \emptyset as the subset of \mathcal{O}. It should be pretty clear that the union of zero sets is the empty set. For the intersection, we’re really applying vacuous truth again. We define x to be in the intersection of \{U_\alpha\} if it is in every U_\alpha, which every x is because there are no U_\alpha for them to not be in. Another way of thinking about it is to observe that \bigcap\{U_1\}\supset\bigcap\{U_1,U_2\}\supset\dotsb for any sequence U_1,U_2,\dotsc. So if we take this chain a step backwards, we get \bigcap\emptyset\supset \bigcap\{U_1\}=U_1 for any U_1.

Oh, and this only works because we’re considering the empty set to be a subset of X. If we just think of a single empty set in the wide world of sets, without thinking of it as “part” of anything, then this same reasoning shows that its intersection is the collection of all sets, which isn’t a set. So either this intersection isn’t defined, or we fiddle with the axioms so that it is defined, or we just ignore it.

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