Topology usually starts with the idea of a metric space. A metric space is a set of values with some concept of distance. We need to define that first, before we can get into anything really interesting.

Metric Spaces and Distance

What does distance mean?

Let’s look at a set, S, consisting of elements s₁, s₂, s₃,…,s_n. What does it mean to measure a distance from s_i to s_j?

We’ll start by looking at a simple number-line, with the set of real numbers. What’s the distance between two numbers x and y? It’s a measure of how far over the number-line you have to go to get from x to y. But that’s really circular; we’ve defined distance as “how far you have to go”, which is defined by distance. Let’s try again. Take a blank ruler, and put it next to the numberline, so that the left edge of the ruler is on X, and then draw a mark on the ruler where it touches Y. The length of the ruler up to that mark is the distance from x to y. The reason that this one isn’t circular is because now, you can take that ruler, and use it to answer the question: is the number v farther from the number w than x is from y? Because the ruler gives you a *metric* that you can use *that is separate from* the number-line itself.

A metric over S is a function that takes two elements s_i and s_j, and returns a *real number* which measure the distance between those two elements. To be a proper metric, it needs to have a set of required properties. To be formal, a function d : S × S → ℜ is a *metric function over S* if/f:

1. ∀ s_i, s_j ∈ S: d(s_i,s_j) = 0 if/f i=j. (the identity property)
2. ∀ s_i, s_j ∈ S: d(s_i,s_j) = d(s_j,s_i) (the symmetry property)
3. ∀ s_i, s_j, s_k ∈ S: d(s_i,s_k) ≤ d(s_i,s_j) + d(s_j,s_j) (the triangle inequality property)
   Some people also add a fourth property, called the non-negativity property; I prefer to leave it out, because it can be inferred from the others. But for completeness, here it is: ∀ s_i, s_j ∈ S: d(s_i,s_j) ≥ 0.
   A metric space is just the pair (S,d) of a set S, and a metric function d over the set.
   For example:
4. The real numbers are a metric space with the ruler-metric function. You can easily verify that properties of a metric function all work with the ruler-metric. In fact, they are are all things that you can easily check with a ruler and a number-line, to see that they work. The function that you’re creating with the ruler is: d(x,y) = |x-y| (the absolute value of x – y). So the ruler-metric distance from 1 to 3 is 2.
5. A cartesian plane is a metric space whose distance function is the euclidean distance:
   d((a_x,a_y), (b_x,b_y)) = ((a_x-b_x)² + (a_y-b_y)² )^1/2.
6. In fact, for every n, the euclidean n-space is a metric space using the euclidean distance.
7. A checkerboard is a metric space if you use the number of kings moves as the distance function.
8. The Manhattan street grid is a metric space where the distance function between two intersections is the sum of the number of horizontal blocks and the number of vertical blocks between them.

Open and Closed Sets in Metric Spaces

You can start moving from metric spaces to topological spaces by looking at open sets. Take a metric space, (S,d), and a point p∈S. An open ball B(p,r) (a ball of radius r around point p) in S is the set of points x such that d(p,x) 0, B(p,r)∩T≠∅. The closure of T (usually written as T with a horizontal line over it; sometimes written as T by computer scientists, because that’s the closure notation in many CS subjects). is the set of all points adherent to T. *(note: a typo was corrected in this paragraph. Thanks to the commenter who caught it!)
A subset T of S is called a closed subset if/f T=T. Intuitively, T is closed if it *contains the surface that forms its boundary. So in 3-space, a solid sphere is a closed space. The contents of the sphere (think of the shape formed by the air in a spherical balloon) is not a closed space; it’s bounded by a surface, but that surface is not part of the space.

The Poincarė conjecture has been in the news lately, with an article in the Science Times today. So I’ve been getting lots of mail from people asking me to explain what the Poincarė conjecture is, and why it’s a big deal lately?
I’m definitely not the best person to ask; the reason for the recent attention to the Poincarė conjecture is deep topology, which is not one of my stronger fields. But I’ll give it my best shot. (It’s actually rather bad timing. I’m planning on starting to write about topology later this week; and since the Poincarė conjecture is specifically about topology, it really wouldn’t have hurt to have introduced some topology first. But that’s how the cookie crumbles, eh?)
So what is it?
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In 1904, the great mathematician Henri Poincarė was studying topology, and came up with an interesting question.
We know that if we look at closed two-dimensional surfaces forming three dimensional shapes (manifolds), that if the three dimensional shape has no holes in it, then it’s possible to transform it by bending, twisting, and stretching – but *without tearing* – into a sphere.
Poincarė wondered about higher dimensions. What about a three dimensional closed surface in a four-dimensional space (a 3-manifold)? Or a closed 4-manifold?
The conjecture, expressed *very* loosely and imprecisely, was that in any number of dimensions *n*, any figure without holes could be reduced to an *n*-dimensional sphere.
It’s trivial to show that that’s true for 2-dimensional surfaces in a three dimensional space; that is, that all closed 2-dimensional surfaces without holes can be transformed without tearing into our familiar sphere (which topologists call a 2-sphere, because it’s got a two dimensional surface).
For surfaces with more than two dimensions, it becomes downright mind-bogglingly difficult. And in fact, it turns out to be *hardest* to prove this for the 3-sphere. Nearly every famous mathematician of the 20th century took a stab at it, and all of them failed. (For example, Whitehead of the infamous Russell & Whitehead “Principia” published an incorrect proof in 1934.)
Why is it so hard?
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Visualizing the shapes of closed 2-manifolds is easy. They form familiar figures in three dimensional space. We can imagine grabbing them, twisting them, stretching them. We can easily visualize almost anything that you can do with a closed two-dimensional surface. So reasoning about them is very natural to us.
But what about a “surface” that is itself three dimensional, forming a figure that takes 4 dimensions. What does it look like? What does *stretching* it mean? What is does a hole in a 4-dimensional shape look like? How can I tell if a particular complicated figure is actually just something tied in knots to make it look complicated, or if it actually has holes in it? What are the possible shapes of things in 4, 5, 6 dimensions?
That’s basically the problem. The math of it is generally expressed rather differently, but what it comes down to is that we don’t have a good intuitive sense of what transformations and what shapes really work in more than three dimensions.
What’s the big deal lately?
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The conjecture was proved for all surfaces with seven or more dimensions in 1960. Five and six dimensions followed only two years later, proven in 1962. It took another 20 years to find a proof for 4 dimensions, which was finally done in 1982. Since 1982, the only open question was the 3- manifold. Was the Poincarė conjecture true for all dimensions?
There’s a million dollar reward for answer to that question with a correct proof; and each of the other proofs of the conjecture for higher dimensions won the mathematical equivalent of the Nobel Prize. So the rewards for figuring out the answer and proving it are enormous.
In 2003, a rather strange reclusive Russian mathematician named Grigory Perelman published a proof of a *stronger* version of the Poincarė conjecture under the incredibly obvious title “The Entropy Formula for the Ricci Flow and Its Geometric Application”.
It’s taken 3 years for people to work through the proof and all of its details in order to verify its correctness. In full detail, it’s over 1000 pages of meticulous mathematical proof, so verifying its correctness is not exactly trivial. But now, three years later, to the best of my knowledge, pretty much everyone is pretty well convinced of its correctness.
So what’s the basic idea of the proof? This is *so* far beyond my capabilities that it’s almost laughable for me to even attempt to explain it, but I’ll give it my best shot.
The Ricci flow is a mathematical transformation which effectively causes a *shrinking* action on a closed metric 3-surface. As it shrinks, it “pinches off” irregularities or kinks in the surface. The basic idea behind the proof is that it shows that the Ricci flow applied to metric 3-surfaces will shrink to a 3-sphere. The open question was about the kinks: will the Ricci flow eliminate all of them? Or are there structures that will *continually* generate kinks, so that the figure never reduces to a 3-sphere?
What Perelman did was show that all of the possible types of kinks in the Ricci flow of a closed metric 3-surface would eventually all disappear into either a 3-sphere, or a 3-surface with a hole.
So now that we’re convinced of the proof, and people are ready to start handing out the prizes, where’s Professor Perelman?
*No one knows*.
He’s a recluse. After the brief burst of fame when he first published his proof, he disappeared into the deep woods in the hinterlands of Russia. The speculation is that he has a cabin back there somewhere, but no one knows. No one knows where to find him, or how to get in touch with him.