# A Great Math Site: Understanding the Analemma

By way of the astronomy picture of the day, I encountered a really fantastic site about the analemma.

The analemma is the apparent path that the sun takes in the sky during the year. If you record the
precise position of the sun at the same time every day, instead of being in exactly the same place
every day, it will traverse a figure eight, like in this image. This is an effect caused by a combination of the eccentricity of the earth’s orbit, and the tilt of the earth’s axis. It can be a bit hard to visualize just where the figure-eight shape comes from; the analemma site uses a combination of diagrams and animations to make it extremely clear, and works through the entire process of demonstrating where each component of the analemma comes from, and deriving the equations that describe it.

# Interesting Parallels: The Leader Election Problem and Notch Receptors

Yesterday at Pharyngula, PZ posted a description of his favorite signaling pathway in developmental biology, the [Notch system.][notch] Notch is
a cellular system for selecting one cell from a collection of essentially indistinguishable cells, so that that one cell can take on some different role.
What I found striking was that the problem that Notch solves is extremely similar to one of the classic problems of distributed systems that we study in computer science, called the leader-election problem; and that
the mechanism used by Notch is remarkably similar to one of the classic
leader-election algorithms, called the [Itai-Rodeh algorithm][ir].
[notch]: http://scienceblogs.com/pharyngula/2006/12/notch.php
[ir]: http://citeseer.ist.psu.edu/itai81symmetry.html
Before I go into detail, I think it’s worth throwing a bit of a dig at
some of the IDist type bozos. This is very much the kind of thing
that IDists look for; they try to find natural systems that strongly
resemble designed systems, so that they can assert that the natural
system must have been designed. But the people allegedly doing ID
research *don’t bother to study the fundamental science*. No one in
ID-land is studying developmental biology, to really understand
the complex signaling pathways, and what I would call the algorithmic
way that they operate. I think it’s quite damning to their arguments
that they don’t study this; but I also think I know *why*. If you
read PZs article, and you starting looking in depth into developmental
pathways, and understanding how they fit together, and how very small
changes in a process can produce drastically different results – well, it
really sort of pulls the carpet out from under the ID argument. You can really see just how these systems evolved in terms of gradual changes. Just look at Notch – how simple the behavior is, how widely it’s used to produce very *different* kinds of differentiation, and how easily the
process can be perturbed to produce different results.

# Complexity from Simplicity; or, Why Casey Luskin Needs a Math Class

One of my fellow ScienceBloggers, [Karmen at Chaotic Utopia](http://scienceblogs.com/chaoticutopia/2006/11/puzzling_at_a_simpleminded_cre.php) pointed out a spectacularly stupid statement in [Casey Luskin’s critique of Carl Zimmer][lutkin] (*another* fellow SBer) at the Discovery Institutes “Center for Science and Culture”. Now normally, I might not pile on to tear-down of Casey (not because he doesn’t deserve it, but because often my SciBlings do such a good job that I have nothing to add); but this time, he’s crossed much too far into *my* territory, and I can’t let that pass without at least a brief comment.
[lutkin]: http://www.evolutionnews.org/2006/11/evolution_nat_geo_1.html
So, here’s the dumb statement:
>The article called evolution a “simple” process. In our experience, does a “simple” process generate
>the type of vast complexity found throughout biology?
Yes, one of the leading IDist writers on the net believes that in reality, simple processes don’t generate complex results.
Karmen pointed out fractals as a beautiful example of the generation of complexity from simplicity. I’d like to point out something that, while lacking the artistic beauty of a well-chosen fractal, is an *even simpler* and possibly more profound example.

# Archimedes Integration of the Circle

A lot of people have asked me to write something about “Archimedes Integration”, and I’m finally getting around to fulfilling that request.
As most of you already know, Archimedes was a philosopher in ancient Greece who, among other things, studied mathematics. He invented a technique for computing areas that’s the closest thing to calculus before Newton and Leibniz. Modern mathematicians call Archimedes technique “the method of exhaustion”.
The basic idea of the method of exhaustion is to take the figure whose area you want to compute, and to divide it into pieces whose area you already know how to compute; and to make the divisions smaller and smaller, *exhausting* the area not included.

# Navier Stokes: False Alarm

There’s bad news on the math front. Penny Smith has *withdrawn* her Navier Stokes paper, because of the discovery of a serious error.
But to be optimistic for a moment, this doesn’t mean that there’s nothing there. Remember that when Andrew Wiles first showed his proof of Fermat’s last theorem, he discovered a very serious error. After that, it took him a couple of years, and some help from a colleague, but he *did* eventually fix the problem and complete the proof.
Whatever develops, it remains true that Professor Smith has made *huge* strides in her work on Navier-Stokes, and if she hasn’t found the solution yet, she has at least helped pave to road to it. Here’s hoping that she finishes it!
Good luck, Professor Smith! We’re pulling for you.

*(Note: in the original version of this, I made an absolutely **huge** error. One of my faults in discussing topology is scrambling when to use forward functions, and when to use inverse functions. Continuity is dependent on properties defined in terms of the *inverse* of the function; I originally wrote it in the other direction. Thanks to commenter Dave Glasser for pointing out my error. I’ll try to be more careful in the future!)*
I’m going to spend a bit more time talking about what continuity means; it’s a really important concept in topology, and I don’t think I did a particularly good job at explaining it in my first attempt.
Continuity is a concept of a certain kind of *smoothness*. In non-topological mathematics, we define continuity with a very straightforward algebraic idea of smoothness. A standard intuitive definition of a *continuous function* in algebra is “a function whose graph can be drawn without lifting your pencil”. The topological idea of continuity is very much the same kind of thing – but since a topological space is just a set with some additional structure, the definition of continuity has to be generalized to the structure of topologies.
The closest we can get to the algebraic intuition is to talk about *neighborhoods*. We’ll define them more precisely in a moment, but first we’ll just talk intuitively. Neighborhoods only exist in topological metric spaces, since they end up being defined in terms of distance; but they’ll give us the intuition that we can build on.
Let’s look at two topological spaces, **S** and **T**, and a function f : **S** → **T** (that is, a function from *points* in **S** to *points* in **T**). What does it mean for f to be continuous? What does *smoothness* mean in this context?
Suppose we’ve got a point, *s* ∈ **S**. Then f(*s*) ∈ **T**. If f is continuous, then for any point p in **T** *close to f(s)*, f-1(p) will be *close to* *s*. What does close to mean? Pick any distance – any *neighborhood* N(f(s)) in **T** – no matter how small; there will be a corresponding neighborhood of M(*s*) around s in **S** so that for all points p in N(f(s)), f-1 will be in M(*s*). If that’s a bit hard to follow, a diagram might help:

To be a bit more precise: let’s define a neighborhood. A neighborhood N(p) of a point p is a set of points that are *close to* p. We’ll leave the precise definition of *close to* open, but you can think of it as being within a real-number distance in a metric space. (*close to* for the sake of continuity is definable for any topological space, but it can be a strange concept of close to.)
The function f is continuous if and only if for all points f(s) ∈ **T**, for all neighborhoods N(f(s)) of f(s), there is some neighborhood M(s) in **S** so that f(M(s)) ⊆ N(f(s)). Note that this is for *all* neighborhoods of *all* points in **T** mapped to by f – so no matter how small you shrink the neighborhood around f(s), the property holds – and it implies that as the neighborhood in **T** shrinks, so does the corresponding neighborhood in **S**, until you reach the single points f(s) and s.
Why does this imply *smoothness*? It means that you can’t find a set of points in the range of f in **T** that are close together, but that weren’t close together in **S** before being mapped by f. f won’t put things together that weren’t together originally. And it won’t pull things apart that weren’t
close together originally. *(This paragraph was corrected to be more clear based on comments from Daniel Martin.)*
For a neat exercise: go back to the category theory articles, where we defined *initial* and *final* objects in a category. There are corresponding notions of *initial* and *final* topologies in a topological space for a set. The definitions are basically the same as in category theory – the arrows from the initial object are the *continuous functions* from the topological space, etc.

# The Genius of Alonzo Church (rerun)

I’m on vacation this week, so I’m posting reruns of some of the better articles from when Goodmath/Badmath was on Blogger. Todays is a combination of two short posts on numbers and control booleans in λ calculus.
So, now, time to move on to doing interesting stuff with lambda calculus. To
start with, for convenience, I’ll introduce a bit of syntactic sugar to let us
name functions. This will make things easier to read as we get to complicated
stuff.
To introduce a *global* function (that is a function that we’ll use throughout our lambda calculus introduction without including its declaration in every expression), we’ll use a definition like the following:
*square ≡ λ x . x × x*
This declares a function named “square”, whose definition is “*λ x . x×x*”. If we had an expression “square 4”, the definition above means that it would effectively be treated as if the expression were: “*(λ square . square 4)(λ x . x×x)*”.
Numbers in Lambda Calculus
——————————
In some of the examples, I used numbers and arithmetic operations. But numbers don’t really exist in lambda calculus; all we really have are functions! So we need to invent some way of creating numbers using functions. Fortunately, Alonzo Church, the genius who invented the lambda calculus worked out how to do that. His version of numbers-as-functions are called Church Numerals.
In Church numerals, all numbers are functions with two parameters:
* Zero ≡ *λ s z . z*
* One ≡ *λ s z . s z*
* Two ≡ *λ s z . s (s z)*
* Three ≡ *λ s z . s (s (s z))*
* Any natural number “n”, is represented by a Church numeral which is a function which applies its first parameter to its second parameter “n” times.
A good way of understanding this is to think of “z” as being a a name for a zero-value, and “s” as a name for a successor function. So zero is a function which just returns the “0” value; one is a function which applies the successor function once to zero; two is a function which applies successor to the successor of zero, etc. It’s just the Peano arithmetic definition of numbers transformed into lambda calculus.
But the really cool thing is what comes next. If we want to do addition, x + y, we need to write a function with four parameters; the two numbers to add; and the “s” and “z” values we want in the resulting number:
add ≡ *λ s z x y . x s (y s z)*
Let’s curry that, to separate the two things that are going on. First, it’s taking two parameters which are the two values we need to add; second, it needs to normalize things so that the two values being added end up sharing the same binding of the zero and successor values.
add_curry ≡ λ x y. (λ s z . (x s (y s z)))
Look at that for a moment; what that says is, to add x and y: create the church numeral “y” using the parameters “s” and “z”. Then **apply x** to that new church numeral y. That is: a number is a function which adds itself to another number.
Let’s look a tad closer, and run through the evaluation of 2 + 3:
add_curry (λ s z . s (s z)) (λ s z . s (s (s z)))
To make things easier, let’s alpha 2 and 3, so that “2” uses “s2” and “z2”, and 3 uses “s3” and “z3”;
add_curry (λ s2 z2 . s2 (s2 z2)) (λ s3 z3 . s3 (s3 (s3 z3)))
Now, let’s do replace “add_curry” with its definition:

(λ x y .(λ s z. (x s y s z))) (λ s2 z2 . s2 (s2 z2)) (λ s3 z3 . s3 (s3 (s3 z3)))
Now, let’s do a beta on add:

λ s z . (λ s2 z2 . s2 (s2 z2)) s (λ s3 z3 . s3 (s3 (s3 z3)) s z)
And now let’s beta the church numeral for three. This basically just “normalizes” three: it replaces the successor and zero function in the definition of three with the successor and zero functions from the parameters to add.
λ s z . (λ s2 z2 . s2 (s2 z2)) s (s (s (s z)))
Now.. Here comes the really neat part. Beta again, this time on the lambda for two. Look at what we’re going to be doing here: two is a function which takes two parameters: a successor function, and zero function. To add two and three, we’re using the successor function from add function; and we’re using the result of evaluating three *as the value of the zero!* for two:
λ s z . s (s (s (s (s z))))
And we have our result: the church numeral for five!
Choice in Lambda Calculus
—————————
Now that we have numbers in our Lambda calculus, there are only two things missing before we can express arbitrary computations: a way of expressing choice, and a way of expressing repetition. So now I’ll talk about booleans and choice; and then next post I’ll explain repetition and recursion.
We’d like to be able to write choices as if/then/else expressions, like we have in most programming languages. Following the basic pattern of the church numerals, where a number is expressed as a function that adds itself to another number, we’ll express true and false values as functions that perform an if-then-else operation on their parameters. These are sometimes called *Church booleans*. (Of course, also invented by Alonzo Church.)
* TRUE ≡ λ t f . t
* FALSE ≡ λ t f . f
So, now we can write an “if” function, whose first parameter is a condition expression, second parameter is the expression to evaluate if the condition was true, and third parameter is the expression to evaluate if the condition is false.
* IfThenElse ≡ *λ cond t f . cond t f*
For the boolean values to be useful, we also need to be able to do the usual logical operations:
* BoolAnd ≡ *λ x y .x y FALSE*
* BoolOr ≡ *λ x y. x TRUE y*
* BoolNot ≡ *λ x . x FALSE TRUE*

Now, let’s just walk through those a bit. Let’s first take a look at BoolAnd. We’ll try evaluating “*BoolAnd TRUE FALSE*”:
* Expand the TRUE and FALSE definitions: “*BoolAnd (λ t f . t) (λ t f . f)*”
* Alpha the true and false: “*BoolAnd (λ tt tf . tt) (λ ft ff . ff)*”
* Now, expand BoolAnd: *(λ t f. t f FALSE) (λ tt tf . tt) (λ ft ff . ff)*
* And beta: *(λ tt tf.tt) (λ ft ff. ff) FALSE*
* Beta again: *(λ xf yf . yf)*
And we have the result: *BoolAnd TRUE FALSE = FALSE*. Now let’s look at “*BoolAnd FALSE TRUE*”:
* *BoolAnd (λ t f . f) (λ t f .t)*
* Alpha: *BoolAnd (λ ft ff . ff) (λ tt tf . tt)*
* Expand BoolAnd: *(λ x y .x y FALSE) (lambda ft ff . ff) (lambda tt tf . tt)*
* Beta: *(λ ft ff . ff) (lambda tt tf . tt) FALSE
* Beta again: FALSE
So *BoolAnd FALSE TRUE = FALSE*. Finally, *BoolAnd TRUE TRUE*:
* Expand the two trues: *BoolAnd (λ t f . t) (λ t f . t)*
* Alpha: *BoolAnd (λ xt xf . xt) (λ yt yf . yt)*
* Expand BoolAnd: *(λ x y . x y FALSE) (λ xt xf . xt) (λ yt yf . yt)*
* Beta: *(λ xt xf . xt) (λ yt yf . yt) FALSE*
* Beta: (λ yt yf . yt)
So *BoolAnd TRUE TRUE = TRUE*.
The other booleans work in much the same way.
So by the genius of Alonzo church, we have *almost* everything we need in order to write programs in λ calculus.

# A Lambda Calculus rerun

I’m on vacation this week, so I’m recycling some posts that I thought were particularly interesting to give you something to read.

In computer science, especially in the field of programming languages, we tend to use one particular calculus a lot: the Lambda calculus. Lambda calculus is also extensively used by logicians studying the nature of computation and the structure of discrete mathematics. Lambda calculus is great for a lot of reasons, among them:

1. It’s very simple.
2. It’s Turing complete.
3. It’s easy to read and write.
4. It’s semantics are strong enough that we can do reasoning from it.
5. It’s got a good solid model.
6. It’s easy to create variants to explore the properties of various alternative ways of structuring computations or semantics.

The ease of reading and writing lambda calculus is a big deal. It’s led to the development of a lot of extremely good programming languages based, to one degree or another, on the lambda calculus: Lisp, ML, and Haskell are very strongly lambda calculus based.

The lambda calculus is based on the concept of *functions*>. In the pure lambda calculus, *everything* is a function; there are no values *at all* except for functions. But we can build up anything we need using functions. Remember back in the early days of this blog, I talked a bit about how to build mathematics? We can build the entire structure of mathematics from nothing but lambda calculus.

So, enough lead-in. Let’s dive in a look at LC. Remember that for a calculus, you need to define two things: the syntax, which describes how valid expressions can be written in the calculus; and a set of rules that allow you to symbolically manipulate the expressions.

### Lambda Calculus Syntax

The lambda calculus has exactly three kinds of expressions:

1. Function definition: a function in lambda calculus is an expression, written: “`λ param . body`” which defines a function with one parameter.
2. Identifier reference: an identifier reference is a name which matches the name of a parameter defined in a function expression enclosing the reference.
3. Function application: applying a function is written by putting the function value in front of its parameter, as in “`x y`” to apply the function `x` to the value `y`.

#### Currying

There’s a trick that we play in lambda calculus: if you look at the definition above, you’ll notice that a function (lambda expression) only takes one parameter. That seems like a very big constraint – how can you even implement addition with only one parameter?

It turns out to be no problem, because of the fact that *functions are values*. So instead of writing a two parameter function, you can write a one parameter function that returns a one parameter function – in the end, it’s effectively the same thing. It’s called currying, after the great logician Haskell Curry.

For example, suppose we wanted to write a function to add x and y. We’d like to write something like: `λ x y . x + y`. The way we do that with one-parameter functions is: we first write a function with one parameter, which returns another function with one parameter.

Adding x plus y becomes writing a one-parameter function with parameter x, which returns another one parameter function which adds x to its parameter:

`λ x . (λ y . x + y)`

Now that we know that adding multiple parameter functions doesn’t *really* add anything but a bit of simplified syntax, we’ll go ahead and use them when it’s convenient.

#### Free vs Bound Identifiers

One important syntactic issue that I haven’t mentioned yet is *closure* or *complete binding*. For a lambda calculus expression to be evaluated, it cannot reference any identifiers that are not *bound*. An identifier is bound if it a parameter in an enclosing lambda expression; if an identifier is *not* bound in any enclosing context, then it is called a *free* variable.

1. *λ x . plus x y*: in this expression, “y” and “plus” are free, because they’re not the parameter of any enclosing lambda expression; x is bound because it’s a parameter of the function definition enclosing the expression “plus x y” where it’s referenced.
2. *λ x y.y x*: in this expression both x and y are bound, because they are parameters of the function definition.
3. *λ y . (λ x . plus x y*: In the inner lambda, “*λ x . plus x y*”, y and plus are free and x is bound. In the full expression, both x and y are bound: x is bound by the inner lambda, and y is bound by the other lambda. “plus” is still free.

We’ll often use “free(x)” to mean the set of identifiers that are free in the expression “x”.

A lambda calculus expression is completely valid only when all of its variables are bound. But when we look at smaller subexpressions of a complex expression, taken out of context, they can have free variables – and making sure that the variables that are free in subexpressions are treated right is very important.

### Lambda Calculus Evaluation Rules

There are only two real rules for evaluating expressions in lambda calculus; they’re called α and beta. α is also called “conversion”, and beta is also called “reduction”.

#### α Conversion

α is a renaming operation; basically it says that the names of variables are unimportant: given any expression in lambda calculus, we can change the name of the parameter to a function as long as we change all free references to it inside the body.

So – for instance, if we had an expression like:

λ x . if (= x 0) then 1 else x^2

We can do α to replace X with Y (written “α[x/y]” and get):

λ y . if (= y 0) then 1 else y^2

Doing α does *not* change the meaning of the expression in any way. But as we’ll see later, it’s important because it gives us a way of doing things like recursion.

#### β Reduction

β reduction is where things get interesting: this single rule is all that’s needed to make the lambda calculus capable of performing *any* computation that can be done by a machine.

Beta basically says that if you have a function application, you can replace it with a copy of the body of the function with references to the parameter identifiers replaced by references to the parameter value in the application. That sounds confusing, but it’s actually pretty easy when you see it in action.

Suppose we have the application expression: “*(λ x . x + 1) 3*“. What beta says is that we can replace the application by taking the body of the function (which is “*x + 1*”); and replacing references to the parameter “*x*” by the value “*3*”; so the result of the beta reduction is “*3 + 1*”.

A slightly more complicated example is the expression:

λ y . (λ x . x + y)) q

It’s an interesting expression, because it’s a lambda expression that when applied, results in another lambda expression: that is, it’s a function that creates functions. When we do beta reduction in this, we’re replacing all references to the parameter “y” with the identifier “q”; so, the result is “*λ x. x+q*”.

One more example, just for the sake of being annoying:

(λ x y. x y) (λ z . z × z) 3“.

That’s a function that takes two parameters, and applies the first one to the second one. When we evaluate that, we replace the parameter “*x*” in the body of the first function with “*λ z . z × z*”; and we replace the parameter “*y*” with “3”, getting: “*(λ z . z × z) 3*”. And we can perform beta on that, getting “*3 × 3*”.

Written formally, beta says:

λ x . B e = B[x := e] if free(e) ⊂ free(B[x := e]

That condition on the end, “if free(e) subset free(B[x := e]” is why we need α: we can only do beta reduction *if* doing it doesn’t create any collisions between bound identifiers and free identifiers: if the identifier “z” is free in “e”, then we need to be sure that the beta-reduction doesn’t make “z” become bound. If there is a name collision between a variable that is bound in “B” and a variable that is free in “e”, then we need to use α to change the identifier names so that they’re different.

As usual, an example will make that clearer: Suppose we have a expression defining a function, “*λ z . (λ x . x+z)*”. Now, suppose we want to apply it:

(λ z . (λ x . x + z)) (x + 2)

In the parameter “*(x + 2)*”, x is free. Now, suppose we break the rule and go ahead and do beta. We’d get “*λ x . x + x + 2*”.

The variable that was *free* in “x + 2” is now bound. Now suppose we apply that function: “*(λ x . x + x + 2) 3*”. By beta, we’d get “3 + 3 + 2”, or 8.

What if we did α the way we were supposed to?

• By α[x/y], we would get “*λ z . (λ y . y + z) (x+2)*”.
• by α[x/y]: `lambda z . (lambda y . y+z)) (x + 2)`. Then by β, we’d get “*λ y . y + x + 2*”. If we apply this function the way we did above, then by β, we’d get “*3+x+2*”.

“*3+x+2*” and “*3+3+2*” are very different results!

And that’s pretty much it. There’s another *optional* rule you can add called η-conversion. η is a rule that adds *extensionality*, which provides a way of expressing equality between functions.

η says that in any lambda expression, I can replace the value “f” with the value “g” if/f for all possible parameter values x, *f x = g x*.

What I’ve described here is Turing complete – a full effective computation system. To make it useful, and see how this can be used to do real stuff, we need to define a bunch of basic functions that allow us to do math, condition tests, recursion, etc. I’ll talk about those in my next post.

We also haven’t defined a model for lambda-calculus yet. (I discussed models here and here.) That’s actually quite an important thing! LC was played with by logicians for several years before they were able to come up with a complete model for it, and it was a matter of great concern that although LC looked correct, the early attempts to define a model for it were failures. After all, remember that if there isn’t a valid model, that means that the results of the system are meaningless!

# Topological Spaces

Yesterday, I introduced the idea of a *metric space*, and then used it to define *open* and *closed* sets in the space. (And of course, being a bozo, I managed to include a typo that made the definition of open sets equivalent to the definition of closed sets. It’s been corrected, but if you’re not familiar with this stuff, you might want to go back and take a look at the corrected version. It’s just replacing a ≤ with a <, but that makes a *big* difference in meaning!)
Today I’m going to explain what a *topological space* is, and what *continuity* means in topology. (For another take on continuity, head on over to [Antopology][antopology] where Ofer has posted his explanation.)
A *topological space* is a set **X** and a collection **T** of subsets of **X**, where the following conditions hold:
1. ∅ ∈ **T** and **X** ∈ **T*.
2. ∀ C ∈ ℘(**T**); : ∪(c ∈ C) ∈ **T**. (That is, the union of any collection of subsets of **T** is an element of **T**. )
3. ∀ s, t ∈ **T** : s ∩ t ∈ T. *(The intersection of any two subsets of **T** is also in **T**.)
The collection **T** is called a *topology* on **T**. The *members* of **T** are the *open sets* of the topology. The *closed sets* are the set complements of the members of **T**. Finally, the *elements* of the topological space **X** are called *points*.
The connection to metric spaces should be pretty obvious. The way we built up open and closed sets over a metric space can be used to produce topologies. The properties we worked out for the open and closed sets are exactly the properties that are *required* of the open and closed sets of the topology. There are many ways to build a topology other than starting with a metric space, but that’s definitely the easiest way.
One of the most important ideas in topology is the notion of *continuity*. In some sense, it’s *the* fundamental abstraction of topology. We can now define it.
A *function* from topological space **X** to topological space **U** is *continuous* if/f for every open sets C ∈ **T** the *inverse image* of f on C is an open set. The inverse image is the set of points x in **X** where f(x) ∈ C.
That’s a bit difficult to grasp. What it’s really capturing is that there are no *gaps* in the function. If there were a gap, then the open spaces would no longer be open. Think of the metric spaces idea of open sets. Imagine an open set with a cube cut out of the middle. It’s definitely not continuous. If you took a function on that open set, and its inverse image was the set with the cube cut out, then the function is not smoothly mapping from the open set to the other topological space. It’s mapping *part of* the open set, leaving a big ugly gap.
Now, here’s were it gets kind of nifty. The set of of topological spaces and continuous functions form a *category*. with the spaces as objects and the functions as morphisms. We call this category **Top**. It’s often easiest to talk about topological spaces using the constructs of category theory.
So, for example, one of the most important ideas in topology is *homeomorphism*. A homeomorphism is a bicontinuous bijection (a bicontinuous function is a continuous function with a continuous inverse; a bijection is a bidirectional total function between sets.) A homeomorphism between topological spaces is a *homomorphism* in **Top**.
From the perspective of topology, any two topological spaces with a homeomorphism between them are *identical*. (Which ends up corresponding exactly to how we defined the idea of *equality* in category theory.)
[antopology]: http://antopology.blogspot.com/2006/08/continuity-introduced.html

# Introducing Topology

Back when GM/BM first moved to ScienceBlogs, we were in the middle of a poll about the next goodmath topic for me to write about. At the time, the vote was narrowly in favor of topology, with graph theory as a very close second.
We’re pretty much done with category theory, so it’s topology time!
So what’s topology about? In some sense, it’s about the fundamental abstraction of *continuity*: if I have a bunch of points that form a continuous line or surface, what does that really mean? In particular, what does it mean *from within* the continuous surface?
Another way of looking at is as the study of what kinds of *structures* are formed from continuous sets of points. This viewpoint makes much of topology look a lot like category theory: a study of mathematical structures, what they mean, and how we can build them and create mappings between them.
Let’s take a quick look at an example. There’s a famous joke about topologists; you can always recognize a topology at breakfast, because they’re the people who can’t tell the difference between their coffee mug and their donut.
It’s not just a joke; there’s a real example hidden in there. From the viewpoint of topology, the coffee mug and the donut *are the same shape*. They’re both toruses. In topology, the exact shape doesn’t matter: what matters is the basic continuities of the surface: what is *connected* to what, and *how* they are connected. In the following diagram, all three shapes are *topologically* identical:

If you turn the coffee mug into clay, you can remold it from mug-shape to donut-shape *without tearing or breaking*. Just squishing and stretching. So in topology, they *are* the same shape. On the other hand, a sphere is different: you can’t turn a donut into a sphere without tearing a whole in it. If you’ve got a sphere and you want to turn it into a torus, you can either flatten it and punch a hole in the middle; or you can roll it into a cylinder, punch holes in the ends to create a tube, and then close the tube into a circle. And you can’t turn a torus into a sphere without tearing it: you need to break the circle of the torus and then close the ends to create a sphere. In either case, you’re tearing at least one whole in what was formerly a continuous surface.
Topology was one of the hottest mathematical topics of the 20th century, and as a result, it naturally has a lot of subfields. A few examples include:
1. **Metric topology**: the study of *distance* in different spaces. The measure of distance and related concepts like angles in different topologies.
2. **Algebraic topology**: the study of topologies using the tools of abstract algebra. In particular, studies of things like how to construct a complex space from simpler ones. Category theory is largely based on concepts that originated in algebraic topology.
3. **Geometric topology**: the study of manifolds and their embeddings. In general, geometric topology looks at lower-dimensional structures, most either two or three dimensional. (A manifold is an abstract space where every point is in a region that appears to be euclidean if you only look at the local neighborhood. But on a larger scale, the euclidean properties may disappear.)
4, **Network topology**: topology in the realm of discrete math. Network topologies are graphs (in the graph theory sense) consisting of nodes and edges.
5. **Differential Topology**: the study of differential equations in topological spaces that have the properties necessary to make calculus work.
Personally, I find metric topology rather dull, and differential topology incomprehensible. Network topology more properly belongs in a discussion of graph theory, which is something I want to write about sometime. So I’ll give you a passing glance at metric topology to see what it’s all about, and algebraic topology is where I’ll spend most of my time.
One of the GM/BM readers, Ofer Ron (aka ParanoidMarvin) is starting a new blog, called [Antopology][antopology] where he’ll be discussing topology, and we’re going to be tag-teaming our way through the introductions. Ofer specializes in geometric topology (knot theory in particular, if I’m not mistaken), so you can get your dose of geometric topology from him.
[antopology]: http://antopology.blogspot.com/