Breakfast at Topology’s

What is topology?

Topology has been, in recent years, one of the most promissing actresses in all Mathwood.

And like all lead actresses, some may love her, some may hate her; but every mathematician has heard of her.

Nice… But what is topology?

To understand it, let’s first be clear about what a continuous transformation is.

Imagine the world is made of play-doh, and you can change the shape of an object the same way you can do it with plastiline.

Now, let’s pretend this clay is so special you can change the color, texture and even the size of any object to your heart’s contempt. A pea can be resized to be bigger than Earth! How awesome does that sound?

But so much power has to come with one condition: no matter how much you change an object, you can never take anything from it, nor add anything to it.

This means that what’s glued together will never be separated, and what’s separated will never be glued together. Good news for your mom: you’ll never be able to break that delicate chinesse porcelain tea pot she loves.

In very raw words, by making this kind of changes on any object, you are applying a continuous transformation to it. So, basically, a continuous transformation is a series of changes that keep apart what used to be apart, and keep together what used to be together.

Mathematicians out there, don’t be upset for I know this is not the formal definition of continuity.

But the picture works, so I’ll work with that picture for now.

Ok. Now that we understand what a continuous transformation is, we can understand what topology is.

Finally! So, what is topology?

Topology is the mathematical study of all those propierties which don’t change after we continuously transform an object.

For example:

Size IS NOT something topology cares about.

Shape IS NOT something topology cares about.

But how many pieces does an object have IS something topology cares about, as well as how many holes does an object have.

So basically, two things are topologically the same if you can continuously transform them into one another.

For example, a cube and a sphere are topologically the same. But not so the sphere and a doughnut (called torus in math).

Mmmm… Doughnuts… Wait! Is there something wrong here?

Let’s see.

As you can verify, a doughnut and a mug are the same for topology!

So if we lived in a world made of topological play-doh, you’d save a lot of money in food. Just imagine ordering a cup of coffee, drinking it, and after that, transform it into a doughnut!

Best breakfast deal ever!