Category Archives: Topology (take 2)

Manifold are the Manifolds

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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

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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.

Multiplying Spaces

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When people talk informally about topology, we always say that the basic idea of equivalence is that two spaces are equivalent if they can be bent, stretched, smushed, or twisted into each other, without tearing or gluing. A mug is the same shape as a donut, because you can make a donut out of clay, and then shape that donut into a mug without tearing, punching holes, or gluing pieces together. A sphere is the same shape as a cube, because if you’ve got a clay sphere, you can easily reshape it into a cube, and vice-versa.

Homeomorphism is the actual formal definition of that sense of equivalence. The intuition is fantastic – it’s one of the best informal description of a difficult formal concept that I know of in math! But it’s not ideal. WHen you take a formal idea and make it informal, you always lose some details.

What we’re going to do here is try to work our way gradually through the idea of transformability and topological equivalence, so that we can really understand it. Before we can do that, we need to be able to talk about what a continuous transformation is. To talk about continuous transformations, we need to be able to talk about some topological ideas called homotopy and isotopy. And to be able to define those, we need to be able to use topological products. (Whew! Nothing is ever easy, is it?) So today’s post is really about topological products!

The easiest way that I can think of to explain the product of two topological spaces is to say that it’s a way of combining the structures of the spaces by adding dimensions. For example, if you start with two spaces each of which is a line segment, the product of those two spaces is a square (or a circle, or an octagon, or …) You started with two one-dimensional spaces, and used them to create a new two-dimensional space. If you start with a circle and a line, the product is a cylinder.

In more formal terms, topological products are a direct extension of cartesian set products. As the mantra goes, topological spaces are just sets with structure, which means that the cartesian product of two topological sets is just the cartesian products of their point-sets, plus a combined structure that preserves attributes of the original structure of the spaces.

Let’s start with a reminder of what the cartesian product of two sets is. Given a set A and a set B, the cartestian product A \times B is defined as the set of all possible pairs (a, b), where a \in A and b \in B. If A=\{1, 2, 3\} and B=\{4, 5\}, then A\times B = \{ (1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5) \}.

In category theory, we take the basic idea of the cartesian product, and extend it to a general product of different mathematical objects. It does this by using the idea of projections. In this model, instead of saying that the product of sets A and B is a set of pairs (a, b), we can instead say that the product is a set S of objects, and two functions P_A : S \rightarrow A and P_B : S \rightarrow B. (To be complete, we’d need to add some conditions, but the idea should be clear from this much.) Given any object in the the product set S, P_A(S) will give us the projection of that object onto A. This becomes more interesting when we consider sets of objects. The A-projection of a collection of points from the product set S is the shadow that those points cast onto the set A.

A topological product is easiest to understand with that categorical approach. The set of points in a product category A \times B is the cartesian product of the sets of points in A and the sets of points in B. The trick, with topologies, is that you need to describe the topological structure of the product set: you need to be able to say what the neighorhoods are. There are lots of ways that you could define the neighborhoods of the product, but we define it as the topological space with the smallest collection of open-sets. To understand how we get that, the projections of the category theoretical approach make it much easier.

Informally, the neighborhoods in the product A \times B are things that cast shadows into the topological spaces A and B which are neighborhoods in A and B.

Suppose we have topological spaces A and B. If S is the product topology A \times B, then it has projection functions P_A: S \rightarrow A and P_B: S \rightarrow P_B.

The projection functions from the product need to maintain the topological structure of the original topologies. That means that the projection function must be continuous. And that, in turn, means that the inverse image of the projection function is an open set. So: for each open set O in A, P_A^{-1}(O) is an open-set in S.

Let’s look at an example. We’ll start with two simple topological spaces – a cartesian plane (2d), and a line (1d). In the plane, the neighborhoods are open circles; in the line, the neighborhoods are open intervals. I’ve illustrated those below.

open-sets

The product of those two is a three dimensional space. The neighborhoods in this space are cylinders. If you use the projection from the product to the plane, you get open circles – the neighborhood structure of the plane. If you use the projection from the product to the line, you get open intervals – the neighborhood structure of the line.

open-set-cyl

One interesting side-point here. One thing that we come across constantly in this kind of formal math is the axiom of choice. The AoC is an annoying bugger, because it varies from being obviously true to being obviously ridiculously false. Topological products is one of the places where it’s obviously true. The axiom choice is equivalent to the statement that given a collection of non-empty topological spaces, the product space is not empty. Obvious, right? But then look at the Banach-Tarski paradox.

Topological Spaces: Defining Shapes by Closeness

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When people talk about what a topological space is, you’ll constantly hear one refrain: it’s just a set with structure!

I’m not a fan of that saying. It’s true, but it just doesn’t feel right to me. What makes a set into a topological space is a relationship between its members. That relationship – closeness – is defined by a structure in the set, but the structure isn’t the point; it’s just a building block that allows us to define the closeness relations.

The way that you define a topological space formally is:

A topological space is a pair (X, T, N), where X is a set of objects, called points; T is a set of subsets of X; and N is a function from elements of X to elements of T (called the neighborhoods of X where the following conditions hold:

  1. Neighborhoods basis: \forall A \in N(p): p \in A: every neighborhood of a point must include that point.
  2. Neigborhood supersets: \forall A \in N(p): \forall B \in X: B \supset A \Rightarrow B \in N(p). If B is a superset of a neighborhood of a point, then B must also be a neighborhood of that point.
  3. Neighborhood intersections: \forall A, B \in N(p): A \cap B \in N(p): the intersection of any two neighborhoods of a point is a neighborhood of that point.
  4. Neighborhood relations: \forall A \in N(x): \exists B \in N(x): \forall b \in B: A \in N(b). If A is a neighborhood of a point p, then there’s another neighborhood B of p, where A is also a neighborhood of every point in B.

The collection of sets T is called a topology on T, and the neighborhood relation is called a neighborhood topology of T.

Like many formal definitions, this is both very precise, and not particularly informative. What the heck does it mean?

In the previous topology post, I talked about metric spaces. Every metric space is a topological space (but not vice-versa), and we can use that to help explain how the set-of-sets T defines a meaningful definition of closeness for a topological space.

In the metric space, we define open balls around each point in the space. Each one forms an open set around the point. For any point p in the metric space, there are a sequence of ever-larger open-balls of points around p.

That sequence of open balls defines the closeness relation in the metric space:

  • a point q is closer to p than it r is if q is in one of the open balls around p, which r isn’t. (In a metric space, that’s equivalent to saying that the distance d(q, p) < d(q, r).)
  • two points q and r are equally close to p if there is no open ball around p where q is included but r isn’t, or where r is included but p isn’t. (In a metric space, that’s equivalent to saying that d(q, p) = d(r, p).)

In a topological space, we don’t neccessarily have a distance metric to define open balls. But the neighborhoods of each point p define the closeness relation in the same way as the open-balls in a metric space!:

  • The neighborhoods N(p) of a point are equivalent to the open balls around p in a metric space.
  • The open sets of the topology (the members of T) are equivalent to the open sets of the metric space.
  • The complements of the members of T are equivalent to the closed sets of the metric space.

One of the most important ideas in topology is the notion of continuity. Some people would say that it’s the fundamental abstraction of topology, because the whole idea of the equivalence between two shapes is that there is a continuous transformation between them. And now that we know what a topological space is, we can define continuity.

Continuity isn’t a property of a topological space, but rather a property of a function between two topological spaces: if (T, X_T, N_T) and (U, X_U, N_U) are both topological spaces, then a function f: X \rightarrow Y is continuous if and only if for every open set C \in X_U, the inverse image of f on C is an open set in X_T. (The inverse image of f is the set of points x \in X_T: f(x) \in C).

Once again, we’re stuck with a very precise definition that’s really hard to make any sense out of. I mean really, the inverse image of the function on an open set is an open set? What the heck does that mean?

What it’s really capturing is that there are no gaps in mapping from one space to the other. If there was a gap, it would create a boundary – there would be a hard edge in the mapping, and so the inverse image would show that as a closed set. 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 only mapping part of the open set, leaving a ugly, hard-edged gap.

In topology, we say that two shapes are equivalent if and only if they can be continuously transformed into each other. In intuitive terms, that continuous transformation means that you can do the transformation without tearing holes are gluing edges. That gives us a clue about how to understand this definition. What the definition means is really saying is pretty much that there’s no gluing or tearing: it says that if a set in the target is an open set, the set of everything that mapped to it is also an open set. That, in turn, means that if f(x) and f(y) are close together in U, then x and y must have been close together in T: so the structure of neighborhood relations is preserved by the function’s mapping.

One continuous map from a topological space isn’t enough for equivalence. It’s possible to create a continuous mapping from one topological space to another when they’re not the same – for example, you could map part of the topology T onto U. As long as for that part, it’s got the continuity properties, that’s fine. For two topologies to be equivalent, there must be a homeomorphism between the sets. That is, a function f such that:

  • f is one-to-one, total, and onto
  • Both f and f^{-1} are continuous.

As a quick aside: here’s one of the places where you can see the roots of category theory in algebraic topology. There’s a very natural category of topological spaces. The objects in the category are, obviously, the topological spaces. The arrows are continuous functions between the spaces. And a homeomorphism (homo-arrow) in the category is a homeomorphism between the objects.

Closeness without distance

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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.

Another pass at Topology!

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A long time ago – in 2006! – I wrote a ton of blog posts about topology. In the course of trying to fix up some of the import glitches from migrating this blog to its new home, I ended up looking at a bunch of them. And… well… those were written in my early days of blogging, and looking back at those posts now… well, let’s just say that my writing has come a long way with 8 years of practice! I was thinking, “I could do a much better job of writing about that now!”

So that’s what I’m going to do. This isn’t going to be reposts,
but rather completely rewrites.

Topology is typical of one of the methods of math that I love: abstraction. What mathematicians do is pick some fundamental concept, focus tightly on it, discarding everything else. In topology, you want to understand shapes. Starting with the basic idea of shape, topology lets us understand shapes, distortions, dimensions, continuity, and more.

The starting point in topology is closeness. You can define what a shape is by describing which points are close to which other points. Two shapes are equivalent if they can be built using the same closeness relationships. That means that if you can take one shape, and pull, squash, and twist it into another shape – as long as you don’t have to either break closeness relations (by tearing or punching holes in it), or add new closeness relations (by gluing edges together) – the two shapes are really the same thing.

This leads to a very, very bad math joke.

How do you recognize topologists at breakfast?

They’re the people who can’t tell their donut from their coffee.

The easiest way to ruin a joke is to over-explain it. Happily, when the joke is this bad, it’s already ruined, so this isn’t my fault. See, in topology, a coffee mug is the same shape as a donut. They’re each three dimensional shapes with one hole. If you had a donut made of an infinitely stretchable material, you could shape it into a coffee mug without every tearing it, ripping a hole, or gluing two edges together.

Anyway – starting tomorrow, I’ll be posting a new version of that old topology series.