Tag Archives: Topology

Manifold are the Manifolds

In the stuff I’ve been writing about topology so far, I’ve been talking about topologies mostly algebraically. I’m not sure why, but for me, algebraic topology is both the most interesting and easy to understand bit of topology. Most people seem to think I’m crazy; in general, people seem to think that algebraic topology is hard. It must say something about me that I find other things much harder, but I’ll leave it to someone else to figure out what.

Despite the fact that I love the algebraic side, there’s a lot of interesting stuff in topology that you need to talk about, which isn’t purely algebraic. Today I’m going to talk about one of the most important ones: manifolds.

The definition we use for topological spaces is really abstract. A topological space is a set of points with a structural relation. You can define that relation either in terms of neighborhoods, or in terms of open sets – the two end up being equivalent. (You can define the open sets in terms of neighborhoods, or the neighborhoods in terms of open sets; they define the same structure and imply each other.)

That abstract definition is wonderful. It lets you talk about lots of different structures using the language and mechanics of topology. But it is very abstract. When we think about a topological space, we’re usually thinking of something much more specific than what’s implied by that definition. The word space has an intuitive meaning for us. We hear it, and we think of shapes and surfaces. Those are properties of many, but not all topological spaces. For example, there are topological spaces where you can have multiple distinct points with identical neighborhoods. That’s definitely not part of what we expect!

The things that we think of as a spaces are really are really a special class of topological spaces, called manifolds.

Informally, a manifold is a topological space where its neighborhoods for a surface that appears to be euclidean if you look at small sections. All euclidean surfaces are manifolds – a two dimensional plane, defined as a topological space, is a manifold. But there are also manifolds that aren’t really euclidean, like a torus, or the surface of a sphere – and they’re the things that make manifolds interesting.

The formal definition is very much like the informal, with a few additions. But before we get there, we need to brush up on some definitions.

  • A set {\mathbf S} is countable if and only if there is a total, onto, one-to-one function from {\mathbf S} to the natural numbers.
  • Given a topological space (T, \tau), a basis \beta for \tau is a collection of open sets from which any open set in \tau can be generated by a finite sequence of unions and intersections of sets in \beta. What this really means is that the structure of the space is regular – it’s got nothing strange like an infinitary-union in its open-sets.
  • A topological space (T, \tau) is called a Hausdorff space if and only if for any two distinct points p, q \in T, there are at least one open set o_p where p \in o_p \land q \not\in o_p, and at least one open set o_q where q \in o_q \land p \not\in o_q. (That is, there is at least one open set that includes p but not q, and one that includes q but not p.) A Hausdorff space just defines another kind of regularity: disjoint points have disjoint neighborhoods.

A topological space (T, \tau) is an n-manifold if/f:

  • \tau has a countable basis.
  • (T, \tau) is a Hausdorff space.
  • Every point in T has a neighborhood homeomorphic to an open euclidean n-ball.

Basically, what this really means is pretty much what I said in the informal definition. In a euclidean n-space, every point has a neighborhood which is shaped like an n-ball, and can be separated from any other point using an n-ball shaped neighborhood of the appropriate size. In a manifold, the neighborhoods around a point look like the euclidean neighborhoods.

If you think of a large enough torus, you can easily imagine that the smaller open 2-balls (disks) around a particular point will look very much like flat disks. In fact, as the torus gets larger, they’ll become virtually indistinguishable from flat euclidean disks. But as you move away from the individual point, and look at the properties of the entire surface, you see that the euclidean properties fail.

Another interesting way of thinking about manifolds is in terms of a construction called charts, and charts will end up being important later.

A chart for an manifold is an invertable map from some euclidean manifold to part of the manifold which preserves the topological structure. If a manifold isn’t euclidean, then there isn’t a single chart for the entire manifold. But we can find a set of overlapping charts so that every point in the manifold is part of at least one chart, and the edges of all of the charts overlap. A set of overlapping charts like that is called an atlas for the manifold, and we will sometimes say that the atlas defines the manifold. For any given manifold, there are many different atlases that can define it. The union of all possible atlases for a manifold, which is the set of all charts that can be mapped onto parts of the manifold is called the maximal atlas for the manifold. The maximal atlas for a manifold is, obviously, unique.

For some manifolds, we can define an atlas consisting of charts with coordinate systems. If we can do that, then we have something wonderful: a topology on which we can do angles, distances, and most importantly, calculus.

Topologists draw a lot of distinctions between different kinds of manifolds; a few interesting examples are:

  • A Reimann manifold is a manifold on which you can meaningfully define angles and distance. (The mechanics of that are complicated and interesting, and I’ll talk about them in a future post.)
  • A differentiable manifold is one on which you can do calculus. (It’s basically a manifold where the atlas has measures, and the measures are compatible in the overlaps.) I probably won’t say much more about them, because the interesting thing about them is analysis, and I stink at analysis.
  • A Lie group is a differentiable manifold with a valid closed product operator between points in the manifold, which is compatible with the smooth structure of the manifold. It’s basically what happens when a differentiable manifold and a group fall in love and have a baby.

We’ll see more about manifolds in future posts!

Squishy Equivalence with Homotopy

In topology, we always talk about the idea of continuous deformation. For example, we say that two spaces are equivalent if you can squish one into the other – if your space was made of clay, you could reshape it into the other just by squishing and molding, without ever tearing or gluing edges.

That’s a really nice intuition. But it’s a very informal intuition. And it suffers from the usual problem with informal intuition: it’s imprecise. There’s a reason why math is formal: because it needs to be! Intuition is great, as far as it goes, but if you really want to be able to understand what a concept means, you need to go beyond just intuition. That’s what math is all about!

We did already talk about what topological equivalence really is, using homeomorphism. But homeomorphism is not the easiest idea, and it’s really hard to see just how it connects back to the idea of continuous deformation.

What we’re going to do in this post is look at a related concept, called homotopy. Homotopy captures the idea of continuous deformation in a formal way, and using it, we can define a form of homotopic equivalence. It’s not quite equivalent to homeomorphism: if two spaces are homeomorphic, they’re always homotopy equivalent; but there are homotopy equivalent spaces that aren’t homeomorphic.

How can we capture the idea of continuous transformation? We’ll start by looking at it in functions: suppose I’ve got two functions, f and g. Both f and g map from points in a topological space A to a topological space B. What does it mean to say that the function f can be continuously transformed to g?

We can do it using a really neat trick. We’ll take the unit interval space – the topological space using the difference metric over the interval from 0 to 1. Call it U = [0, 1].

f can be continuously deformed into g if, and only if, there is a continuous function t: A \times U \rightarrow B, where \forall a \in A: t(a, 0) = f(a) \land t(a, 1) = g(a).

If that’s true, then we say t is a homotopy between f and g, and that f and g are homotopic.

That’s just the first step. Homotopy, the way we just defined it, doesn’t say anything about topological spaces. We’ve got two spaces, but we’re not looking at how to transform one space into the other; we’re just looking at functions that map between the spaces. Homotopy says when two functions between two spaces are loosely equivalent, because one can be continuously deformed into the other.

To get from there to the idea of transformability of spaces, we need to think about what we’re trying to say. We want to say that a space A can be transformed into a space BB. What does that really mean?

One way to say it would be that if I’ve got A, I can mush it into a shape B, and then much it back to A, without ever tearing or gluing anything. Putting that in terms of functions instead of squishies, that means that there’s a continous function f from A to B, and then a continous function g back from B to A. It’s not enough just to have that pair of functions: if you apply f to map A to B, and then apply g to map back, you need to get back something that’s indistinguishable from what you started with.

Formally, if A and B are topological spaces, and f: A \rightarrow B and g: B \rightarrow A are continuous functions, then the spaces A and B are homotopically equivalent – equivalent over squishing and remolding, but not tearing or gluing – if f \circ g is homotopic with the id function on A, and g \circ f is homotopic with the id function on B.

That captures exactly the notion of continuous transformation that we tried to get with the intuition at the start. Only now it’s complete and precise – we’ve gotten rid of the fuzziness of intuition.

Closeness without distance

In my introduction, I said that topology is fundamentally built on the notion of closeness. Someone very quickly responded on Twitter, because they thought that was wrong. It wasn’t wrong, but it’s easy to see where the confusion came from. Like so much math, Topology is built on a very precise logical and set-theoretic formalism. Mathematicians build those formalisms not because they’re annoying people who want to be mysterious and incomprehensible, but because the precision of those formalisms is critically important.

When you hear a statement like “point A is close to point B in a space S”, you have an intuitive idea of what the word “close” means. But when you try to expand that to math, it could potentially mean several different things. The easiest meaning would be: the distance between A and B is small.

Mathematicians have used that definition for a lot of interesting work. It’s got one limitation though: For it to work, you need to be able to define “distance” in the space. How do you do that? In conventional Euclidean space, we have an easy definition. Describe the position of the two points using Cartesian coordinates: A=(x1, y1), B = (x2, y2). The distance between A and B is:

d(A, B) = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

But we’re moving towards the world of topology. We can’t count on our spaces to be Euclidean. In fact, the whole point of topology is, in some sense, to figure out what happens when you have different spatial structures – that is, structures other than the familiar Euclidean one! We need to be able to talk about distances in some more general way. To do that, we’ll create a new kind of space – a space with an associated distance metric. This new space is called a metric space.

A distance metric is conceptually simple. It’s just a special kind of function, from pairs of points in a space to a real number. To be a distance metric, it needs a couple of properties. Suppose that the set of points in the space is S. Then a function d: S \times S \rightarrow \mathbf{R} is a distance metric if it satisfies the following requirements:

  1. Identity: \forall s_i, s_j \in S: d(s_i, s_j) = 0 \Leftrightarrow s_i = s_j
  2. Symmetry:\forall s_i, s_j \in S: d(s_i, s_j) = d(s_j, s_i)
  3. Triangle Inequality: \forall s_i, s_j, s_k \in S: d(s_i, s_k) \le d(s_i, s_j) + d(s_j, s_k)
  4. Non-negativity: \forall s_i, s_j \in S: d(s_i, s_j) \ge 0

A metric space is just the pair (S,d) of a set S, and a metric function d over the set. For example:

  1. A cartesian plane is a metric space whose metric function is the euclidean distance: d((a_x,a_y), (b_x,b_y)) = \sqrt{(a_x-b_x)^2 + (a_y-b-y)^2}.
  2. A checkerboard is a metric space with the number of kings moves as the metric function.
  3. 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.

All of this is the mathematical work necessary to take one intuitive notion of closeness – the idea of “two points are close if there’s a small distance between them” and turn it into something formal, general, and unambiguous. But we still haven’t gotten to what closeness means in topology! It’s not based on any idea of distance. There are many topological spaces which aren’t metric spaces – that is, there’s no way to define a metric function!

Fortunately, metric spaces give us a good starting point. In topological spaces, closeness is defined in terms of neighborhoods and open balls.

Take a metric space, (S, d), and a point p \in S. An open ball B(p, r) (that is, a ball of radius r around the point p) is the set of points x \in S | d(p, x) < r.

Given a large enough set of points, you can create an infinite series of concentric open spheres: B(p, \epsilon), B(p, 2\epsilon), B(p, 3\epsilon), and so on. Once you’ve got that series of ever-smaller and ever-larger open balls around a point p, you’ve got another notion of closeness. A is closer to p than B is if A is in a smaller open ball around p.

This is the heart of topology. You can define something like an open-ball on a set without a metric. As long as you can create a consistent sequence of open balls, where each larger ball is a strict superset of all of the smaller ones, you can define closeness without any notion of measurable distance!

In the next post, we’ll use this notion of a distance-free sense of closeness to define what a topology actually is.

The Banach-Tarski non-Paradox

For some reason, lately I’ve been seeing a bunch of mentions of Banach Tarski. B-T is a fascinating case of both how counter-intuitive math can be, and also how profoundly people can misunderstand things.

For those who aren’t familiar, Banach-Tarski refers to a topological/measure theory paradox. There are several variations on it, all of which are equivalent.

The simplest one is this: Suppose you have a sphere. You can take that sphere, and slice it into a finite number of pieces. Then you can take those pieces, and re-assemble them so that, without any gaps, you now have two spheres of the exact same size as the original.

Alternatively, it can be formulated so that you can take a sphere, slice it into a finite number of pieces, and then re-assemble those pieces into a bigger sphere.

This sure as heck seems wrong. It’s been cited as a reason to reject the axiom of choice, because the proof that you can do this relies on choice. It’s been cited by crackpots like EE Escultura as a reason for rejecting the theory of real numbers. And there are lots of attempts to explain why it works. For example, there’s one here that tries to explain it in terms of density. There’s a very cool visualization of it here, which tries to make sense of it by showing it in the hyperbolic plane. Personally, most of the attempts to explain it intuitively drive me crazy. One one level, intuitively, it doesn’t, and can’t make sense. But on another level, it’s actually pretty simple. No matter how hard you try, you’re never going to make the idea of turning a finite-sized object into a larger finite-sized object make sense. But on another level, once you think about infinite sets – well, it’s no problem.

The thing is, when you think about it carefully, it’s not really all that odd. It’s counterintuitive, but it’s not nearly as crazy as it sounds. What you need to remember is that we’re talking about a mathematical sphere – that is, an infinite collection of points in a space with a particular set of topological and measure relations.

Here’s an equivalent thing, which is a bit simpler to think about:

Take a line segment. How many points are in it? It’s infinite. So, from that infinite set, remove an infinite set of points. How many points are left? It’s still infinite. Now you’ve got two infinite sets of the same size. So, now you can use one of the sets to create the original line segment, and you can use the second one to create a second, identical line segment.

Still counterintuitive, but slightly easier.

How about this? Take the set of all natural numbers. Divide it into two sets: the set of even naturals, and the set of odd naturals. Now you have two infinite sets,
the set {0, 2, 4, 6, 8, …}, and the set {1, 3, 5, 7, 9, …}. The size of both of those sets is the ω – which is also the size of the original set you started with.

Now take the set of even numbers, and map it so that for any given value i, f(i) = i/2. Now you’ve got a copy of the set of natural numbers. Take the set of odd naturals, and map them with g(i) = (i-1)/2. Now you’ve got a second copy of the set of natural numbers. So you’ve created two identical copies of the set of natural numbers out of the original set of natural numbers.

The problem with Banach-Tarski is that we tend to think of it less in mathematical terms, and more in concrete terms. It’s often described as something like “You can slice up an orange, and then re-assemble it into two identical oranges“. Or “you can cut a baseball into pieces, and re-assemble it into a basketball.” Those are both obviously ridiculous. But they’re ridiculous because they violate one of our instinct that derives from the conservation of mass. You can’t turn one apple into two apples: there’s only a specific, finite amount of stuff in an apple, and you can’t turn it into two apples that are identical to the original.

But math doesn’t have to follow conservation of mass in that way. A sphere doesn’t have a mass. It’s just an uncountably infinite set of points with a particular collection of topological relationship and geometric relationships.

Going further down that path: Banach-Tarski relies deeply of the axiom of choice. The “pieces” that you cut have non-measurable volume. You’re “cutting” from the collection of points in the sphere in a way that requires you to make an uncountably infinite number of distinct “cuts” to produce each piece. It’s effectively a geometric version of “take every other real number, and put them into separate sets”. On that level, because you can’t actually do anything like that, it’s impossible and ridiculous. But you need to remember: we aren’t talking about apples or baseballs. We’re talking about sets. The “slices” in B-T aren’t something you can cut with a knife – they’re infinitely subdivided, not-contiguous pieces. Nothing in the real world has that property, and no real-world process has the ability to cut like that. But we’re not talking about the real world; we’re talking about abstractions. And on the level of abstractions, it’s no stranger than creating two copies of the set of real numbers.

Topoi Prerequisites: an Intro to Pre-Sheafs

I’m in the process of changing jobs. As a result of that, I’ve actually got some time between leaving the old, and starting the new. So I’ve been trying to look into Topoi. Topoi are, basically, an alternative formulation of mathematical logic. In most common presentations of logic, set theory is used as the underlying mathematical basis – set theory and a mathematical logic built alongside it provide a complete foundational structure for mathematics.

Topoi is a different approach. Instead of starting with set theory and a logic with set theoretic semantics, Topoi starts with categories. (I’ve done a bunch of writing about categories before: see the archives for my category theory posts.)

Reading about Topoi is rough going. The references I’ve found so far are seriously rough going. So instead of diving right in, I’m going to take a couple of steps back, to some of the foundational material that I think helps make it easier to see where the category theory is coming from. (As a general statement, I find that category theory is fascinating, but it’s so abstract that you really need to do some work to ground it in a way that makes sense. Even then, it’s not easy to grasp, but it’s worth the effort!)

A lot of category theoretic concepts originated in algebraic topology. Topoi follows that – one of its foundational concepts is related to the topological idea of a sheaf. So we’re going to start by looking at what a sheaf is.

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