The Fundamentals of Topology dealt with less math and more intuition

Introduction:

Ever thought that a cup and a donut are similar – Yeah, me neither (though I dunk donuts in cups of coffee regularly). However, it turns out that these two seemingly unrelated bodies share many properties in common. Let’s talk about that.

In school, you must have heard that circles are just special cases of ellipses. How’s that so? Well for the math geeks here, you could easily prove that using the equations of their shape, but let’s think of it in another way – Deformation.
A circle is just a compressed version of a larger ellipse, you can compress any ellipse into a circle (Fig 1) or you can stretch any circle into an ellipse. When you compress an ellipse you get an infinite number of more ellipses, a circle is just one of them, a special compression. If this thought tickles your curiosity, think about this: if circles are just a special case of ellipses, can it mean that there are some properties which are preserved during these “deformations”.

Figure 1: An ellipse (Purple) can be compressed into infinite number of ellipses which includes circles (the ones coloured black and red). It can also be thought that the central circle (red) is stretched to form larger ellipses or in fact it can also be compressed to form smaller ellipses (not shown in the figure)

What might be the properties that both an ellipse and circle have in common? The answer is surprisingly simple: No matter how the ellipses are deformed, they are still conic sections. Conic sections are basically the sections formed in a cone when a plane cuts through it, this encompasses all of the basic curves we know: Circles, Parabolas, Hyperbolas and Ellipses.

Figure 2: The Conic sections. We can also see here, how a circle is a special case of an ellipse. A conic section becomes a circle if and only if the plane that cuts through the cone is parallel to the base of the cone and also does not pass through the base. But for a conic section to be an ellipse it just has to not pass through the base of the cone, it doesn’t necessarily need to be parallel to the base.
Courtesy: Magister_Mathematicae, CC BY-SA 3.0, Link

If you expected some other property, like maybe a special point being equidistant from something on the curve, it turns out that it actually isn’t the case. The only property that carries over is that they are conic sections. Since a parabola or a hyperbola are also conic sections, does that mean that they can be deformed into one another and into ellipses. Yes, they can easily be deformed into one another for example in a parabola take the two bottom points (the points touching the base of the cone) now just slowly bring them closer. You will notice that the final shape formed when both the points touch is, you guessed it, an ellipse. So any conic section can be deformed into one another and the only property that is preserved is that they still are conic sections even though they are deformed.

Let us go back to the initial question: Are a cup and a donut equivalent? It turns out that the donut can be deformed into a cup, you probably wont believe me so see this well known Gif:

Figure 3: A Donut being morphed into a cup and vice versa

Isn’t that Gif trippy. It is very interesting to see how a cup and a donut, formally called a torus, are morphed into one another. Let’s see another example of this deformation, a cow into a sphere.

Figure 4: A cow being morphed into a Sphere
Courtesy: Keenan Crane, CC BY-SA 4.0, Link

When three dimensional objects are deformed to form other three dimensional objects it begs the question what properties of one are transferred over to other and what objects other than a sphere for example, can be morphed into a cow. All these questions are answered by Topology, the study of properties of an object which are preserved through deformations.

Before we dive deep into a few basic concepts of topology, I have to address a small note, topology is the study of objects when they are deformed i.e stretching, bending, crumpling, twisting but we cannot tear something away from the object or glue something to the object.

A Topological Space:

A Topological space when thought in the form of sets has this definition:
A Topological space is a set (X, T) where T is a collection of subsets of X such that:

1. The empty set “{ }” and the set “X” are present in the set T.
2. The intersection of any number of sets in T must be present in T.
3. The Union of any number of sets must be present in T.
T is called a “topology on X”.

Take a moment to understand these rules, you might not understand them completely, but you will after seeing a few examples.
X = {-2, -1, 0, 1, 2}, can T be :
1. { { }, {-2, -1, 0, 1, 2} } , yes T can be such a set as it obeys all the three rules stated above this is also called a trivial set as it only contains the empty set and X, any topology on X other than this builds upon this set as a framework adding more sets to this while still obeying the rules if this basic framework is absent it is not a topology.
2. { { }, {-2}, {-2, -1, 0, 1, 2} }, yes check it yourself
3. { { }, {-2}, {-1}, {-2, -1, 0, 1, 2} }, no T cannot be this set as the union {-1,-2} is absent
4. { { }, {-2,-1,0}, { 0, 1, 2}, {-2, -1, 0, 1, 2} }, no T cannot be this set, though the union between {-2,-1,0} and { 0, 1, 2} is present (namely {-2, -1, 0, 1, 2} ). This is because the intersection between {-2,-1,0} and { 0, 1, 2}: {0} is absent in the set this can be called a topology only if that set is present.

After understanding this slightly tricky definition of a topological space the majority of you will ask what is the use of all this mathematical jargon? Well, this mathematical jargon is used to fuel another more convoluted but a lot more interesting concept which is the life and soul of topology – homeomorphism.

Homeomorphism:

Remember I told you that a circle can be morphed into an ellipse, well it can even be morphed into a square, a rectangle, a parallelogram etc. So what is the underlying law that governs what shapes a circle can take, or for that matter any 3 dimensional shape, when deformed. What is the relation between the deformed shape and the original shape ? It is Homeomorphism. When a shape is morphed into a new shape, then the new shape and the original shape are called Homeomorphic. So, it follows that if any two shapes, which can be seemingly unrelated are homeomorphic, then one of those two shapes can be deformed into another.

So how can we say if two functions are homeomorphic:
Let us say there are topological spaces X and Y, if there exists a function f: X →Y such that:

  1. Function f is a bijection
  2. Function f is continuous
  3. Function f-1 , i.e the inverse function is also continuous.

These rules might seem a little complicated if you are not familiar with the terms used, but it is actually simple, let’s look at each of these rules carefully. Before we do that we need to revise what a function is – it is basically like a box in which you input a variable, in this case it is X and you get an output, Y.
Rule one says a function is a bijection which essentially means that every input you give to the function is mapped to a unique output in such a way that there are no inputs which aren’t mapped to an output and vice versa. Basically no unpaired elements.
As for Rule 2 it is very easy it just means that the graph of the function is continuous with no breaks.
Rule 3 is also simple it is essentially saying that when the input is Y and the output is X even in that case the graph of the function should be continuous just to make sure that continuity holds from both sides.

Figure 5: An example of a bijective mapping

Let us look at a few examples of homeomorphic spaces:

  1. Any interval, open or closed, for example (a1, a2) is homeomorphic to the set of real numbers R. How so? Because there are infinite real numbers between a1and a2 and there are also infinite real numbers in the set of real numbers so you can take a bijective mapping in such a way that you map the first number of (a1, a2), “a1” to the first number in the set of real numbers i.e “-∞” and you can continue this mapping till you map a2 to infinity, remember here we use the fact that there are infinite numbers between any two real numbers.
  2. Another example as I have mentioned earlier is homeomorphism between a unit square and a unit disc one can be deformed into another. There exists a bijective function between them.

There are lot more examples of homeomorphic objects, but their bijective functions and the analyses for each one of them is out of reach of this blog as it is involves a little bit of complex math which is what I am trying to avoid. But if one is interested in learning them there are countless resources available online. Why I brought up homeomorphism, even though it is a little complicated, is because, as I have already mentioned, it is the life and soul of topology. It is the only way you can start comparing whether two objects can be morphed into each other, let alone their properties. This is just the beginning of Topology, there are a lot more intricate concepts embossed in it, which is why it is one of the most happening places in research of mathematics. But, no need to fear we will try our best to understand as many of those concepts intuitively as we can without delving deep into the mathematics.

An interesting homeomorphism:

Figure 6: The featured image of this blog: A Trefoil knot
Courtesy: Wiebew – CC BY-SA 3.0, Wikimedia Commons


Look at figure 6, it is formally called a Trefoil knot. It is actually a very beautiful image just look at it closely for a while and you will notice that it is a complete structure with no breaks. Now look at a another figure, a torus:

Figure 7: A Torus
Courtesy: Leonid 2, Wikimedia Commons

Would you believe me if I said the trefoil know in figure 6 and the torus in figure 7 are homeomorphic and can be morphed into one another. I think you wouldn’t cause I didn’t believe it either in fact before we discuss this fact in detail, do you think a rectangular sheet would be homeomorphic to a torus and in turn to a trefoil knot. You would surely say “Abhiram! are you crazy, how can a rectangle and a trefoil know be morphed into one another! The Torus has a hole how in the world can you morph a sheet into a torus!”. Hold your horses for a minute an look at this Gif below:

Figure 8: A Rectangular sheet being morphed into a torus.

It’s fascinating isn’t it, how easily a sheet can be bent into a torus. That simple gif, kind of humbled me. It changed my opinion on what I thought could be morphed into what.
When I read about concepts like this I always think, that Maths is so beautiful, especially when understood intuitively.

Before we go back to the torus knot problem, I want to introduce a term –isotopy. It is fundamentally a path connecting two homeomorphic objects while in turn passing through many other homeomorphisms. So basically, if we take figure 8 for example, when the rectangular sheet is being bent into a cylinder all the objects in between those two shapes are homeomorphic and when the cylinder is being bent to a torus all the shapes in between are also homeomorphic. So, the path you chose to make a sheet of paper into a torus is the isotopy between them, there are many kinds of isotopies in topology but they wont be deal here. Now, let us select the isotopy of figure 8 since it is flowing through an infinite number of shapes the only way you can represent them is a function and like I said earlier, the math of this wont be dealt here. If you see the figure 1 even, the big ellipse (purple) when compressed into the smaller circle (red) you can clearly see the homeomorphic ellipses in between and the isotopy of the conversion.

A Torus Knot:

Since we have already veered away from toruses and knots I would like to introduce an interesting shape, it is also very famous in the world of mathematics – A Torus knot. To construct it, all you have to do is to take a string and continuously loop it around the surface of a torus.

Figure 9 : A torus knot, you can hopefully see a torus shape around which the knot is made.

Let us see the knot in other angles:

Figure 10: Top view of the torus It looks very beautiful in this angle
Figure 11: A Point of view which makes the knot look like a bunch of garbage.

The torus knot is very beautiful, in fact there are so many other kinds of knots in the world of math that there is a whole website dedicated to it : Knot-plot they even have an app for plotting knots and it is dope. I would even like to share an image from their website:

Figure 12: An image consisting of 120 Different Torus-Knots! And they are just torus knots imagine the other types of knots possible, if you want. to actually see many many more knots like this I highly suggest to take a look at www.knotplot.com and download their app to create more knots like these. Disclaimer: Knot_Plot has not sponsored me I just like their website a lot and I just wanted to share it.
Link to Original Image Copyright © 1998-2000 by Robert G. Scharein
Figure 13: A torus knot made on a bagel, it is a funny post and I suggest you check it out:
How to make a mathematically correct breakfast by George Hart

Understanding torus knots is very essential to see how trefoil knots and toruses are linked to each other. Even though trefoil knots and torus knots are completely different knots mathematically they are both still very interesting and beautiful to look at.

Shall we finally answer the question that I have been dodging so far: Homeomorphism between Trefoil Knots and toruses. Well, to convert a Trefoil knot into a torus we have to un-knot it. How do we do that? It is complicated and I felt that if I tried to explain how it is unknotted, I wouldn’t be doing justice to the concept as I am not very familiar with it myself. So I have decided to share a post dedicated to just that: Unknotting the Trefoil
Just bear in mind that he draws the trefoil in a different way than what we discussed but if you look closely you will notice that they are the same.

Conclusion:
You might have noticed that we actually didn’t cover a lot of concepts in this blog for instance there is no detailed mention about what properties are preserved when an object is morphed into another, the main focus of topology. But, we did cover homeomorphism at least up to a basic level. There are so many more interesting topics in topology that I want to cover, the link of topology to quantum field theory or, how the Königsberg Problem is the beginning of topology or, the different types of topology and their specific rules or, materials in which topological rules are broken etc. etc. Trust me, I will cover each and every one of them in my upcoming blogs but until then, be happy that you learnt something new today.

Take Home Message:

A topologist cannot tell the difference between a donut and a coffee mug.

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