Wednesday, September 27, 2006

Topological Spaces and Open Sets

In the previous post, when I talked about continuity, I based this on the notion of a point being near to a set. We said that a point p was near to a set A if every neighborhood of the point intersected A, and we said that in our case neighborhoods on the real line were open intervals of the form (a,b), where this is all points x such that a < x < b. We then said that a function f was continuous if for any set A and point p near to it, f(p) is near to f(A)={f(a) a Î A}.

Well, this should come as no surprise to mathematicians looking at this blog, but I was lying then. Neighborhoods can be much more complicated than just open intervals, and in fact this complexity is what makes topology interesting and difficult. We can either approach this by first studying metric spaces and deducing topology from there, as Mark did on Good Math, Bad Math,or start directly from the abstract notion of topological spaces, which is what i'll do.

Basically, a topological space X is just a set of elements, which we shall call points from now on,with a specified family U={Ua}a ÎJ of subsets of X, called the open sets of X (the J is an indexing set, as explained in the next paragraph).

Before we continue, we need a small side trip. Topologies are usually interesting only when the number of points and the number of open sets are uncountable. To be able to work with those sets, we can't use any countable set of indices (such as the naturals), and therefore we use an indexing set J which is simply some set of unspecified cardinality.

The open sets U of X must fulfill some axioms:

  1. X and the empty set Æ must be open.
  2. Any union of open sets must be open. Formally, this means that if we have a subset J¢ Í J of the indexing set, then Èa ÎJ¢ Ua Î U.
  3. Any finite intersection of open sets must be open. Formally, this means that if we have a subset {j1,¼,jn} Í J of the indexing set, then Çi=1nUji Î U.
We can now formulate a formal condition for a set being open. A set U Í X is open if and only if for each point p ÎU there is an open set Up such that p Î UpÍ U. Informally, a set U is open if and only if each of its points has a "small enough" open set around it, that is an open set which is a subset of U.

We can prove this easily, in what looks like cheating. If U is open, then for every point p we can take Up=U, and thus have the necessary open set.

For the other direction, we notice that we can write U=Èp Î U Up. At first glance, this looks very weird. p ÎUp for every p Î U, and so we definitely have U ÌÈp Î UUp. However, it seems impossible that ÈpÎ UUp ÌU, since it would seem that taking all the points and the sets around them would give us a set bigger then the original set U. Here the "small enough" demand comes into play. Since each of the sets Up sits completely inside U, their union must also sit completely inside U. To finish off, we remember that each of the Upare open by assumption, and since an arbitrary union of open sets is open, this means that U=ÈpÎ UUp is open.

This little tautology gives us the connection between the definitions of nearness and open sets. We can reformulate the notion of a point being near to a set precisely. A point p is near to a set A if for every open set U such that p ÎU, we have UÇA being nonempty.

This means that a set U is open if and only if non of its points are near to its complement X-U. For if p Î U,and U is open, there is an open set Up such that p Î Upand Up sits completely inside U, that is Up Ì U. Thus, of course,Up does not intersect the complement of U. Conversely, if none of the points in U are near to X-U, this means that every point p Î U has a"small enough" open set around it which does not intersect X-U, and we can again show that U is the union of all these "small enough" open sets and thus it is open.

This fact makes open sets the natural setting for questions of continuity, or any other concept formulated by nearness.

As a little exercise, I write down this claim, which is usually given as the definition of continuity, and the proof will come next time.

Proposition 1 A function f:X® Y between two topological spaces is continuous if and only for any set U Ì Y which is open, the set f-1(U)={xÎ X f(x) ÎU} is open in X.

Wednesday, August 23, 2006

Continuity Introduced

hen explaining the basic definitions of topology, it is always hard deciding whether to start with Wthe abstract definition, or with a concrete example from which we can derive the axioms. While the abstract definition requires the least amount of background information, it is the least illuminating, and so we will start with the concrete examples encountered when the subject was at its infancy, and slowly derive the basic definitions.

Topologies on a set and the maps preserving them

Basically, a topological space is simple a set, whose elements we will call points from now on, with an additional structure, the topology, put on it. The same set might have many different possible topologies. The study of topology deals with the classification of those different topologies.

We've talked about the fact that in topology we classify spaces, where two spaces are equivalent if we can get from one to the other by stretching and squeezing. However, the notion of stretching and squeezing is of course no basis for a rigorous mathematical theory, and therefore we need to formalize this notion. It is convenient to think about what this means for any two topological spaces, and not only equivalent ones. This means that we essentially ask what kind of mappings we can have between any two topological spaces, which respect the topological structure (the same way that we want linear maps to preserve vector space structure). These are the continous maps. Forgetting the formal definition for now, the essence of this is the fact that we want maps that send points which are near in the domain to points which are near in the range.

A Concrete Example - Two Maps on the Real Line

The best way to study the notion of near points is to look at the space most familiar to us, the real line (actually, when we talk about the real line, we already assume a topology for it). Topology deals with continuous manipulations of the topological spaces, and continuity is relatively easy to describe for functions on the real line. In school math we played a lot with such functions, by drawing graphs for them on a 2-dimensional page. Informally, such a graph is continuous if we can draw it without lifting our pen from the paper. For example, the simplest parabola f(x) =x2 is clearly continuous.

It is also easy to give an example of a non-continuous function. We define a function g(x) which is zero everywhere except at the point 0, where we set g(0)=1. It is impossible to draw a graph for g(x) without lifting our pen from the paper at 0.

Let's try to understand the difference between the two cases, and from it try to get a definition of continuity. For f(x)=x2, every two points close to each other on the real line, are sent to two points close to each other on the graph. However, for g(x) this is no longer true, since the point 0 is sent to a point which is at a distance of 1 at least from the points close to it.

This definition is not that good for the real line, or for almost any space, because of the imprecision in defining what we mean by two points close to each other. Since for any two real numbers we have infinitely many real numbers between them, it makes no sense to talk about the number next to a number p. Instead, we establish a relationship between points on the line and sets of points on the line.

We now need some notations. An open interval on the line, denoted by (a,b) for two real numbers a,b such that a < b, is the set of all points x such that
a < x < b. Given a point p on the line, a neighborhood of p is an interval (a,b) such that a < p < b.

We can finally define nearness. We say that a point p is near to a set A if every neighborhood of p has a point in common with A. For example, we can look at the point 0 and the point (0,1). 0 is near to the interval (0,1), since every neighborhood of 0 is of the form (a,b) for a < 0,b > 0 and this sort of neighborhood has many points in common with (0,1) (to be precise, the entire set of points (0,Min(1,b))).

Having defined near, we can finally talk about continuous functions in a more formal manner. How do we turn the intuitive definition into a mathematically rigorous one? One way to do it is to say that a function f is continuous if for any point p and set A, f(p) is near to the set f(A)={f(a)a Î A}.

Using this definition, let's look at g(x) again. We now see that this function clearly violates the definition. Previously, we saw that 0 was near to (0,1) on the real line. However g(0)=1 but g(x)=0 for all x ¹ 0. This means that we need to ask if g(0)=1 is near to the set g((0,1))={0}, and it's clear that, for example, the neighborhood (1/2,2) of g(0)=1 has no point in common with (0,1).

To show that f(x)=x2 is continuous takes some more work, but simple techniques from Calculus 1 suffice.

Those familiar with the definition of continuity for real functions might ask where has the dreaded e,d definition have run off to. The answer is simple. They are the same definition in the case of the real line. The advantage of the definition I'm using here is the fact that it generalizes to spaces where there is no notion of distance, and thus looking at all points close to a point has no meaning.

We can now look at continuity in the case of the real line, and see what's the only intrinsic information coming from the real line that we used in the definition. The point appearing in the definition is arbitrary, and so is the set we are checking nearness to. The only thing that we took as a given was the fact that a neighborhood of a point is an open interval. In the next posts we look at the abstract definition of neighborhood, and see that this completely defines the topology of the space.

File translated from
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On 16 Aug 2006, 12:43.

Welcome to Antopology

What is this blog all about?

I plan to write a series of blog entries which are intended to be an introduction to topology and geometry, aimed at people with very basic mathematical backgrounds. To the cognisenti, I add the disclaimer that the eventual target of this blog is a user friendly explanation of concepts in differential, algebraic, geometric, and combinatorial topology,which are much more my cup of tea, but I shall start with the basics of point-set topology. To all the rest, have no fear, I plan to take my time and very slowly work through the basic concepts before reaching my goal. Now, let’s proceed to the show.

What is topology all about?

Before we start going into the deeper math, I’ll spend some time trying to show some of the basic questions asked in topology. Topology deals with the study and classification of abstract spaces. Roughly, topology deals with properties of spaces, which can’t be changed by stretching the space or squeezing it. This is best illustrated by the well known mathematical joke that to a topologist a coffee mug and a bagel are one and the same, since they both have one hole (I said that this was a well known joke, never said it was a good one). To a topologist, the fact that the mug has one hole, its handle, and the bagel has also only one hole, means that they are the same space topologically speaking.

Life as an Ant on a Line

The abstraction above is one of the hardest concepts to grasp, and therefore we will start with more concrete examples we can visualize or simply draw. Take a piece of paper and draw a line on it, which might curve around, but doesn’t intersect itself at any point. We want to study properties of this line. In order to dothis, we shall take the ant’s eye view. Imagine an ant living on this line, being unable to leave it or even look outside this line. It can go forwards or backwards, but that’s it. Now let’s give our ant a little car, and send her on a trip on the line. For the sake of our discussion, we refer to this line as the road from now on.

We can draw the road in many ways. We can make it straight and long, or straight and short. We can curve it nicely, or we can draw a very squiggly line. We can even give it“corners”, meaning we draw it with right angle turns in it, or any other angles we wish. As long as we never intersect ourselves while we draw, or lift our pencil from the paper, we are dealing with the same topological space. All we care about in topology is“relations between points”, meaning which point is near to which other point. Roughly, if our ant is standing at point A, and point B is“next” to point A (this is a gross over-simplification, since on our line there are an infinite number of points between A and B) any manipulation of the line which leaves its topology unchanged would have to leave B “next” to A.

Intuitively, we are much more geometric creatures, as is our ant. While topologically we can mess the road up as much as we want (again, as long as we don’t lift our pen or intersect ourselves), the ant will feel differently about each change in the road. Let us imagine that our ant lives on one end of the line, and frequents a pub at the other end. As long as the line is straight, the ant can get home even when it is drunk, since there is nowhere she can fall of the line. Now change the road to a curved road. The ant would have to drive much more carefully, since the road curves and she might have an accident along the way. Now we can do something nastier, by drawing the road with a right angle in it. The topology of this road is unchanged, but almost any attempt to drive on it will end in an accident. When our ant reaches the right angle, she’ll drive off the edge of her world. The poor ant has to come to a complete stop, turn the car to the right, and resume driving.

Basically, we already have some questions we can try to answer. How many topologically different roads can we have, and how do we know that those are all our options? What happens if we change the world that the ant lives on, to one that is two-dimensional (meaning that to the ant her world looks like the surface of a plane, much as the globe looks to humans, as has already been explored by E. Abbot in “Flatland”. See “The Annotated Flatland: A Romance of Many Dimensions” by E. Abbot and I. Stewart, 2001.) How many such worlds exist? How do we tell them apart?

The aim of topology is basically the classification of spaces. We will be exploring the techniques for this classification in detail.

The more established math blogger MarkCC of Good Math, Bad Math will be tag teaming with me on this subject.