The purpose of this note is to give a basic idea of how to study the topology of spaces, which is a mathematical way to describe the shape of an object. To do this, we will concentrate on examples of two-dimensional spaces, or surfaces. In particular, we will explain why they are said to have two dimensions; then we will introduce some basic tools of topology, and show how to use them to tell our different examples apart.

Two-dimensional spaces

The Sphere

For us, a sphere will denote the surface of a round object, such as a billiard ball or the Earth (since it isn't really flat, a huge globe will do). Imagine someone moving on the surface of this sphere, say a sailor on an ocean. Now for various reasons he may want to know his position, which means to be able to describe it in a non-ambiguous way. The standard procedure (at least in the navy) is to calculate the two coordinates known as latitude and longitude.

Now latitude and longitude are said to be independent in the following sense: it means our sailor can pick arbitrarily any value for a longitude and any value for a latitude¹, then go to the location defined by these two values. Incidentally, this explains why the surface of a planet is a 'two-dimensional' space: you have a system of two independent coordinates describing your position on it.

The Plane

A plane is a flat surface stretching indefinitely in all directions, like a huge sheet of paper. Now how would someone describe an ant's position on a plane? The easiest way to do it is first to draw two intersecting (usually perpendicular) straight lines on the plane and then to determine the ant's projections onto those axes. Again, the two resulting coordinates are independent; in some sense, the location of an ant on a plane can be described by two independent points on two lines (or, equivalently, by two real numbers).

The Torus

A torus is a surface that is shaped like the surface of a (ring) doughnut or a tyre tube. With a little practise² it can be readily seen that a nice choice of coordinates on a torus is given by two independent points on two circle, the trick being to consider cuts of the torus by means of horizontal planes and vertical planes. Once again, this means that a torus is intuitively a two-dimensional object.

Let's see what this means. First, we have to settle terminology. A plane will be declared horizontal if it is perpendicular to the symmetry axis of the torus which goes through the hole (thus not touching the torus), and vertical if it contains this axis. Now choose a point on your torus. Cut the torus with a vertical plane containing this point. You'll get two circles (the intersections of the torus and the plane), one of which will contain the given point: the diagram consisting of this circle together with the point on it will give you the first coordinate. Similarly, cut the torus with an horizontal plane containing the given point: again, you'll get two circles (or just one circle, for two specific choices of your horizontal plane), this time having same center. One of these will contain the given point: the diagram consiting of this circle together with the point on it will give you the second coordinate.

A nice alternative description arises from some old video games. Suppose you have a character who can move in any direction on a computer screen, with the property that if he vanishes on the right (or on the left) he appears again on the left (or on the right) at the same horizontal level, and if he vanishes on the top (or on the bottom), he reappears on the bottom (or on the top) at the same vertical level. Then the character can be seen as actually moving on a torus. One way to see this is take a sheet of paper the size of your computer screen, then glue the edges together on the points where the character can appear and disappear. It means you have to take a piece of paper and roll it so the two longer edges meet forming a tube, then link the two ends of the tube to form the torus.

The Klein Bottle³

A Klein bottle is another example of two-dimensional space, whith curious properties somewhat analog to those of a Möbius strip (a one-sided ribbon). One can construct in in two equivalent ways, similar to those used for the torus.

The first way goes like this. Start with a bendy and hollow tube. Assume its two extremities have lids which, to make description easier, should be painted blue on the outside and red on the inside. Then you have to bend the tube and glue the lids together. If you do it the natural way (the two blue sides together), you get a torus, which is not what we want. To get a Klein bottle, you have to glue the blue side of one lid to the red side of the other lid.

This procedure seems to be impossible, since one part of the tube has to go through another part. Well, this is perfectly right; the model is said to have self-intersection. In reality, the Klein bottle shouldn't intersect itself, but then it would mean building it inside a four-dimensional space, an even greater difficulty to overcome. Nevertheless, you can draw nice pictures of this model; some examples can be found here. Note that this model possibly inspired the term 'bottle'.

The other way to realise a Klein bottle is to create a computer-based simulation of someone who walks on its surface. To do this, take a computer screen on which you can move a small character, first allowing him to move anywhere on it, but never to leave it. From his point of view, this would mean the same as moving on a sheet of paper. Then, as for the torus, you have to allow the character to go through the upper and lower sides of the screen, reappearing on the opposite side at the same vertical level. From the character's point of view, it amounts to moving on a screen bent in such a way that the upper side meets the lower side to form a tube.

Now for the tricky part: we want to allow the character to move through the left and right sides of the screen so that he will appear to be moving on a Klein bottle. So, assume the character moves out of the screen on the left or right side. To take the bottle's weird twist into account, you have to make him reappear on the opposite side, but on the line that joins his leaving point and the center of the screen. Alternatively, you can measure the distance between his vanishing point and the top of the screen and take his reappearing point to be on the opposite side, at the same distance measured from the bottom of the screen.

A few experiments will show that, when going through the left or right sides of the screen, the character's right and left sides will be swapped. For instance, to see a two-dimensional bi-coloured crab walking on a Klein bottle, you may follow this link.

Sphere, Plane, Torus and Klein Bottle are Different

A Matter of Shapes

Well, it is quite obvious that a sphere, a plane, a torus and a Klein bottle are all different from each other! In fact, it is not so easy to prove, and depends on what we mean by 'different'. Experts say that the four considered spaces are not homeomorphic to each other. Roughly speaking, two spaces are homeomorphic⁴ if you can reshape one into the other by stretching, twisting, compressing and dilating it, but not tearing it⁵.

For instance, a soccer ball and a rugby ball are homeomorphic: sit on the first one and you get the second one. A punctured sphere (that is, a sphere with a little hole in it -one point missing) is homeomorphic to a plane: let's explain how to see this. Turning your sphere you may assume the North pole is removed. Now replace it with a lamp; then lay down a plane below the South pole. Now, all points lying on the sphere will have one and only one shadow (a projection) on the plane below; conversely all points on the plane will be the shadow of some (uniquely defined) point on the sphere. Using this trick allows one to see how to flatten and stretch out a punctured sphere into a plane, moving all point of the sphere to its projection on the plane. We leave as an exercise for the reader to see that a doughnut and a coffee mug are homeomorphic.

Now, one may want to be able to tell whether two spaces are homeomorphic or not. It's usually very hard, since one cannot really try all possible deformations of one space and expecting never to recognise the other one! So many tools have been introduced to do so. The idea is to attach a property (or a number or whatever) to a space that would be common to homeomorphic spaces, in other words, which is invariant with respect to homeomorphisms. For example, if a space is made out of several pieces, then the number of pieces is an invariant.

Orientability

The first invariant we'll introduce concerns orientation of a two-dimensional space. A surface is non-orientable if it contains a Möbius strip, ie if you can cut out a bit that is (homeomorphic to) a Möbius strip. A more visual way to express it it to say such a surface has only one side, in other words, if you cut a door in your space, you need not open it to get on the other side, all you have to do is walk around⁶.

Now it is obvious that a sphere, a plane and a torus are all orientable: a sheet of paper has two sides, whereas a sphere and a torus have an inside and an outside. Now the Klein bottle is not orientable: you can try and use a picture of the self-intersecting model to imagine an ant going from one side of an imaginary door to the other side just walking around. On the other hand, you may want to find a Möbius strip as part of a Klein bottle: this is quite easily done if you consider the computer-screen model. If you cut out an horizontal stripe extending on the whole breadth of the screen, all you have to do is to check that the way the right and left sides are connected make this stripe into a one-sided ribbon: this is left as an exercise. What we have shown is that a Klein bottle is homeomorphic to none of the three other considered spaces (sphere, plane, torus).

Loops and Simple Connectedness

The invariant we shall introduce now is based on the notion of loops. Fix a point in a given space and call it 'home'. Starting from there, take a walk -any walk- on the given space and eventually come back home. So as not to forget how you walked around, leave a rope on your way: the rope should leave a picture of the path you walked. This path, or the figure drawn by the rope, is called a loop.

Now take one end of the rope and pull it until not possible anymore. If you can get back the whole of the rope, then the path you've walked along is null, in some sense (experts say isotopic to zero). Now call a space simply connected if all loops are isotopic to zero. Now this is an invariant property of spaces. To see this, all one has to check is that the notions of 'loops' and 'pulling a rope' are invariants with respect to homeomorphisms, which is more or less intuitive (but quite messy to write down properly). For instance, if you deform your space without tearing it, your rope will get deformed as well but will still look like a rope (you won't flatten it, nor will you break it); besides, its starting and ending points, being the same before deformation, will remain the same after deformation, so the loop remains a loop.

Loops on a Sphere, a Plane, a Torus

Now, it is easy to see intuitively that a plane and a sphere are simply connected. Let's see why this is not the case for a torus: pull a string through the hole of a doughnut, so that the doughnut will hang to it, then leave the string to rest on the doughnut's surface. It is obviously a loop on the torus, but no matter how tight you pull, you'll always have a piece of doughnut trapped in it (it will always be hanging). Hence what we have shown is that a torus is isomorphic neither to a sphere nor to a plane.

All we are left to see is that a sphere and a plane are not homeomorphic. To do so, we will make a (somewhat involved) proof by contradiction. So we make the assumption that a sphere and a plane are homeomorphic, and, from this, we shall derive an absurd conclusion.

Recall that a punctured sphere (or plane) is just a sphere (or plane) with one single point missing; it is obvious that the resulting space does not depend on which specific point you remove⁷. Now, if a sphere and a plane were homeomorphic, then a punctured sphere and a punctured plane would be homeomorphic as well. For, if you can deform a sphere into a plane, performing that very same deformation on a sphere with one point missing will eventually give you a plane on which one point is missing, namely the point located on the spot onto which the initial point should have moved.

Now we have seen previously that a punctured sphere is homeomorphic to a full plane, so this space is simply connected. On the other hand, a punctured plane is not simply connected: to see this, drive a spike in the plane on the location of the removed point and twist a rope around it. Then, no matter how tight you pull the rope, never shall it go through the spike; in other words, there will always remain a little bit of string lying on the punctured plane. Such a loop is by definition not isotopic to zero, and hence, a punctured plane is not simply connected.

What we have shown is that, should a plane and a sphere be homeomorphic, then there would exist two homeomorphic spaces, one of which being simply connected and the other not: this is a contradiction, so our hypothesis has to be false. Therefore, a plane and a sphere are not homeomorphic.