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.









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On 16 Aug 2006, 12:43.

2 Comments:

Blogger ScienceDave said...

Good show- please keep it up. I had intro to real analysis and topology around a decade and a half ago, and my field (chemistry) gives me little opportunity to stay sharp.

Your first post was good stuff, too, and I think you need not apologize to mathematicians or hardcore math phys people in your audience. They are the outliers that find arid abstraction lip-smacking good.

The rest of us 'run of the mill' scientists, and to an even greater extent, total laymen, will never take the time to see through the thicket of rigor to the beauty deep inside mathematics without a lot of examples and other support.

I understand the need for rigor, but I also see the corrosive influence of the Bourbaki school forbidding "mathematical graven images" repelling the greater communtiy of scientists who find abstraction counter to the way that we work- we build models that we know are wrong and misleading, to get better ideas, and to ultimately construct better, more abstract and more accurate models. These same scientists, when someone comes along willing to bridge the gap between the mathematical rigor and the beauty, will often become quite mathophillic, and seek to apply all they can.

I would never have thought to minor in math, but for one prof who was quick with an example to motivate abstractions. Later, when I used group theory in quantum mechanics and inorganic chemistry, the motivations and mechanisms of using all the fantastic math was made so much easier by having been led, adroitly, through the rigor.

So you are doing important stuff, a sort of mathematical evangelism that could be truly helpful to the science community.

4:39 PM  
Blogger Paranoid Marvin said...

Thanks for your support!

Sorry for taking so long to answer, long vacation and much longer writer's block. I plan to post agian in the next few days.

12:56 PM  

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