Topology vs "a" Topology

[MUSIC PLAYING] We've talked about topology here in Infinite Series.

It's the branch of math where we study the properties of shapes that are preserved no matter how you bend, twist, stretch, or deform them.

And you've probably come across some cool examples of the shapes or topological spaces, like spheres and tori, Mobius bands and Klein bottles.

But what is a topological space, really?

If you were to look in a textbook, you'd find that the definition of a topological space looks something like this.

A topological space starts with a set X and some collection of subsets of X, that satisfy three axioms.

But what are those axioms?

And what do they mean?

And what in the world does this have to do with stretchy shapes?

Today, I'd like to unwind the definition.

We'll do it in a fun, but imperfect way.

We'll use an analogy that's hopefully intuitive, but it won't address all of the nuances of A topology.

My goal is just to get you used to the ideas, and I've put a few links in the description, if you'd like to dive deeper.

To start, did you know that there is a difference between topology and A topology?

The article A is very important.

Topology without an A before it is that branch of math that study shapes or topological spaces.

A topology, on the other hand, is different.

A topology is an axiomatic way to construct a topological space.

It's what we'd see in a textbook, and it's short for the phrase, A topology on a set, which helps us describe how points in a set relate to each other in terms of space.

Believe it or not, all of the cool topological spaces you hear about, have their origins in these basic axioms of A topology on a set.

So what are they?

Instead of telling you the answer, which we could just read out of a book, let me give you some motivation.

A while ago, Kelsey showed us in the episode, "When Pi is Not 3.14," that there are lots of ways to measure distance between points on an XY plane.

The Euclidean metric is the most familiar, but it's just one of many others, like the taxi cab metric and the LP metrics.

And if you thought generalizing the Euclidean metric was cool, you're going to love what comes next.

We can actually generalize the idea behind a metric itself.

In the episode, "When Pi is Not 3.14," we saw that the set of all points at a distance one from the origin, forms a circle in the Euclidean metric, a diamond in the taxi cab metric, and square-type shapes in the LP metrics.

And if we consider points whose distance is less than but not equal to one, we get a circle, diamond, and square that are filled in, without their boundaries.

These shapes are called open because they don't include the boundary points.

But what if you want to make sense of distance between things that aren't numbers?

And what if you want to make sense of nearness and farness without using numbers?

This is where topology comes in.

A topology is an axiomatic way to generalize the open shapes you get when you have a metric.

So what is A topology exactly?

To discover the answer, let's think about something simple, like the set of real numbers-- the number line.

And let's say, that my height is some number x. I won't tell you what x is, but suppose you'd like to verify if it's bigger than 64 inches, but smaller than 67 inches.

Does x lie in the open interval, 64, 67?

Let's call this interval I.

To help you determine the answer, I'm going to give you a machine.

Here's how the machine works.

You can feed any number into it, like my height, and the machine will analyze that number.

If the number is in between, but not equal to 64 and 67, then the machine will eventually terminate.

That's the machine's way of verifying, yes, Tai-Danae is somewhere in between 5 foot 4 and 5 foot 7.

But it's a funny machine.

If you feed in a number that is not in between 64 and 67, then the machine will keep running forever and ever, and never stop.

So that's how you can figure out if my height is in a certain range.

Just guess a number and feed that number into the machine.

If your guess is in between 64 and 67, the machine terminates.

If not, the machine keeps running.

But you might be wondering, why not include the end points, 64 and 67?

Won't the machine terminate if we input the number 64?

Well, think about it.

If you feed 64 into the machine, you're asking the question, is Tai-Danae exactly 64 inches tall?

But there's no way you can ever be sure that I am exactly 64.0000, and so on, inches tall.

There is no ruler with that much precision.

The endpoints 64 and 67 are not verifiable so let's not include them.

Because the machine terminates when you input any number that lies in the interval, I, let's call the machine an indicator machine.

It indicates, or verifies, when a number is in I.

And since every number in I is verifiable by the machine-- in other words, every number in I causes the machine to terminate-- let's just call the interval, I, a verifiable set.

It's precisely the set of all real numbers that cause its indicator machine to terminate.

Now what happens if there are a bunch of machines, where each one is tuned in to a different set?

Let's also call these sets verifiable, if the same property holds.

A set, U, is verifiable if it has an indicator machine that terminates when you input a number that is in U.

And it runs indefinitely, when you input a number that's not in U.

In other words, we can verify when a number is an element of U.

Here's where it gets interesting.

What happens if we have infinitely many verifiable sets?

U1, U2, U3, and so on.

Is their union still verifiable?

And number x is in the union of the sets U1 and U2.

If x is either in U1 or U2, or possibly both.

So if we have infinitely many of these sets, can we verify when a number lies in U1 or in U2, or U3, and so on?

Absolutely.

We just have to plug that number into each of their indicator machines.

If we run the machines in parallel and wait long enough, eventually, one of those machines will terminate.

Hence, the union of verifiable sets is verifiable.

That's observation number one.

Now what if we take the intersection of these sets?

Is the resulting set verifiable?

A number x is in the intersection of U1 and U2, if it lies in both U1 and U2.

So if we have infinitely many of these sets, can we verify when a number lies in U1, and U2, and U3, and so on?

To do that, we'd have to input a number into each machine and wait for all of them to terminate.

But this would take an infinite amount of time.

The process would never end.

But what if we intersect only finitely many of them?

Is the result verifiable this time?

Sure.

We just have to wait for all the machines to start running.

This may take a while, but we can verify that they will all terminate eventually.

So the finite intersection of verifiable sets is verifiable.

This is observation number two.

Finally, suppose we have an indicator machine for the entire set of real numbers.

Then, the whole set is itself, verifiable since every real number is a real number.

And the empty set, the set that contains no numbers, is also verifiable.

Try pausing the video to think about why.

This is observation number three.

Pretty sensible observations, right?

Number one, any union of verifiable sets is verifiable.

Number two, a finite intersection of verifiable sets is verifiable.

And number three, the empty set and the real line itself are verifiable.

But this is exactly what it means to have A topology on the set of real numbers.

A topology is simply a collection of verifiable subsets.

Except, we've been using the word verifiable for illustrative purposes.

The standard word is open, since as we saw earlier, murky things happen on the boundary of open intervals.

Let's put it all together.

A topology on the set of real numbers are, is a collection of subsets called open sets, that satisfy three axioms.

Any union of open sets is open.

The intersection of finitely open sets is open.

And R and the empty set are open.

More generally, we can replace R by any set X, and this defines A topology on the set X.

And this is the definition we've been waiting for.

A topology on X is a collection of subsets, satisfying these three axioms.

By the way, it's possible to define many different topologies on a set, but once you choose one, that set together with its topology is called a topological space.

What's cool is that A topology, these abstract open sets, are the right generalization of the open shapes-- the circles, diamonds, and squares-- that come from the Euclidean, taxi cab, and LP metrics.

So in addition to doing math with numbers, these axioms give us just the right foundation to do abstract mathematics with shapes, like Mobius bands and Klein bottles and their deformations.

I'll close by leaving you with one example of a topological space, which we can verify using the three axioms.

Ready?

Here it is.

How is this a topological space?

For the solution, check out the link to "Mathema" in the description below.

There, you'll also find this week's challenge problem.

Once you take a look at the problem, think about the answer and shoot me an email at pbsinfiniteseries@gmail.com, with the subject line, Point Set Challenge.

A correct solution will be chosen at random, to receive a PBS Digital Studios t-shirt.

See you soon.