Understanding Compactness and Projective Reals: A Beginner's Guide

[marellasunny]

Can someone please explain 'compactness' by using 'projective reals'?
I understand compactness.I can't quite grasp the term 'projective reals' and make the connection.Please help!I'm a beginner.
(Side note:wiki calls it as a 'the one-point compactification of the real line, or the projectively extended real numbers').Thank you.

 

[micromass]

The projective reals actually have to do with projective geometry. But explaining projective geometry would probably take us too far as a simpler explanation is possible.

We can define the projective reals by taking the ordinary real numbers $\mathbb{R}$ and adding a special element $\infty$. This element is intuitively both negative and positive infinity. So we can imagine traversing the real line and once we arrive at positive infinity, we will start back from the very beginning. So the space is "looped" in some sense.

Of course, this has nothing to do with compactness. I can't even talk about compactness now, since I haven't even defined some kind of metric (or topology) on the projective reals. How do we define a metric structure?? Note that the projective reals remind us a lot of a circle, so we might want to use that analogy.

We might want to embed $\mathbb{R}$ in the circle, this can be done by "stereographic projection". Here is a figure:

Basically, if we have a real number x, then we draw a line between (x,0) and (0,1). The place where that line intersects the circle is the stereographic projection.

If we do the calculations, then we get

$$x\rightarrow (\frac{2x}{x^2+1},\frac{x^2-1}{x^2+1})$$

So right now we can see $\mathbb{R}$ as a part of the circle. In particular, we can identify $\mathbb{R}$ with the circle without the north pole (as the north pole is the only point not in the image of stereographic projection). So adjoining $\infty$ with the reals, is the same as adjoining (0,1) with the rest of the circle. So the circle is a model of the projective reals. The good thing here is that the circle has a natural metric structure/topology, so compactness makes sense. And indeed, the circle is closed and bounded and thus compact!

 

[Bacle2]

The projective space are (can be seen as ) a parametrization of the collection all lines

going thru the origin, i.e., the collection of 1-dimensional subspaces of a space. Projective

spaces can be defined over every field. Still, for the reals, you consider all lines going

through the origin. But every line thru (0,0) in R^2* can be identified uniquely by a point it

goes thru. We can then choose, for convenience the points in standard S¹ that

a line goes thru. But then, we have that if a line goes thru p=(x,y), it also goes thru the

antipode of p given by (-x,-y). So we consider just one of these 2; e.g., we consider the

upper part of S¹. But even then, (1,0) and (-1,0) are equivalen-- since a line

going thru (1,0) also goes thru (-1,0). So our projective real space can be seen as the

upper-half of S¹ with (1,0) and (-1,0) identified. This is just a topological S

¹.

As to compactness, you can show this using the fact that projective line is a quotient

space, i.e., it is the space S¹/~ , where (x,y)~(x',y') iff (def.) x'=-x, y'=-y.

And quotient maps are continuous by construction. Then the projective line is the

quotient---continuous image --of a compact space , so it is compact.


There are generalizations; not only can we define projective line, plane, but we can also define PRⁿ, and PKⁿ , for K a field, by generalizing the concept of a line, plane, etc. in Kⁿ, as the set of (resp.) multiples λk of some k in Kⁿ, or as the collection of combinations λx+ty for a plane, etc. Of course, for fields other than ℝ , geometric interpretations are harder.




* From now on, line means line thru (0,0).


And then the concept can be extended to that of the Grassmanian, which parametrizes subspaces of different dimensions. The Grassmanian is also a manifold.

 

[mathwonk]

in a compact space a sequence never "runs off to infinity", i.e. all sequences have limit points.

in a given non compact space one way to achieve this is to look at sequences which do go off to infinity and add in some points to be limit points of such sequences, i.e. to add in some points (that were previously) "at infinity".

This can be done to essentially any space, and in many ways. One of the simplest ways is the "one point compactification", achieved by adding in only one point at infinity.

There is another way that works on euclidean space, "projectification". In every dimension n, the real projective space P^n is a certain nice compactification of euclidean space R^n.

When n = 1, the two methods agree and P^1 happens to be the one point compactification of R^1. In general P^n is realizable as the quotient space of the n sphere, by identifying antipodal points.

The original space R^n is thought of as represented by say the northern hemisphere, and the new points added in are the pairs of antipodal points on the equator. (The equator of a circle has only one such pair of points.)

 

[Bacle2]

in a compact space a sequence never "runs off to infinity", i.e. all sequences have limit points.
This can be done to essentially any space, and in many ways. One of the simplest ways is the "one point compactification", achieved by adding in only one point at infinity.

There is another way that works on euclidean space, "projectification". In every dimension n, the real projective space P^n is a certain nice compactification of euclidean space R^n.

When n = 1, the two methods agree and P^1 happens to be the one point compactification of R^1. In general P^n is realizable as the quotient space of the n sphere, by identifying antipodal points.

The original space R^n is thought of as represented by say the northern hemisphere, and the new points added in are the pairs of antipodal points on the equator. (The equator of a circle has only one such pair of points.)

I think you need your space to be locally-compact and Hausdorff to have a 1-pt compactification, or maybe to have the compactified space be Haudorff.

Edit: I was wrong here; we can always have a 1-pt. compactification by adding an extra point to X , so that X* :=X\/{p}, and giving X* a topology , with

open sets those in X , together with the complements of closed, compact sets:

http://en.wikipedia.org/wiki/Alexandroff_extension.

 

[Bacle2]

You may say that compactness is a second best to finiteness. When you have a finite collection, a bunch of nice things happen, like having actual max, min values, and spaces are simple. Compactness gives you, beside what Wonk said-- we need the Wonk, give us the Wonk--e.g., for metric spaces, every sequence has a convergent subsequence.

The study of compactness becomes kind of specialized with different types and perspectives: countable compactness, sequential compactness, limit-point compactness, etc.

 

[mathwonk]

thanks for fixing my sloppiness. this seemed to be a question that could benefit from some general intuitive answers.

 

[Bacle2]

Hopefully both the rigor and the intuition will combine for a better understanding. Maybe when I reach your level I can lower my guard and be less rigorous, but I must first get the rigor to then see the bigger picture.