In mathematics, a topological space $X$ is called countably generated if the topology of $X$ is determined by the countable sets in a similar way as the topology of a sequential space (or a Fréchet space) is determined by the convergent sequences.
The countably generated spaces are precisely the spaces having countable tightness—therefore the name countably tight is used as well.
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The Banach–Tarski Paradox

Laurent Bartholdi  Imbeddings in groups of subexponential growth

Mod01 Lec03 Sigma Algebra generated by a class
Transcription
Hey, Vsauce. Michael here. There's a famous way to seemingly create chocolate out of nothing. Maybe you've seen it before. This chocolate bar is 4 squares by 8 squares, but if you cut it like this and then like this and finally like this you can rearrange the pieces like so and wind up with the same 4 by 8 bar but with a leftover piece, apparently created out of thin air. There's a popular animation of this illusion as well. I call it an illusion because it's just that. Fake. In reality, the final bar is a bit smaller. It contains this much less chocolate. Each square along the cut is shorter than it was in the original, but the cut makes it difficult to notice right away. The animation is extra misleading, because it tries to cover up its deception. The lost height of each square is surreptitiously added in while the piece moves to make it hard to notice. I mean, come on, obviously you cannot cut up a chocolate bar and rearrange the pieces into more than you started with. Or can you? One of the strangest theorems in modern mathematics is the BanachTarski paradox. It proves that there is, in fact, a way to take an object and separate it into 5 different pieces. And then, with those five pieces, simply rearrange them. No stretching required into two exact copies of the original item. Same density, same size, same everything. Seriously. To dive into the mind blow that it is and the way it fundamentally questions math and ourselves, we have to start by asking a few questions. First, what is infinity? A number? I mean, it's nowhere on the number line, but we often say things like there's an infinite "number" of blahblahblah. And as far as we know, infinity could be real. The universe may be infinite in size and flat, extending out for ever and ever without end, beyond even the part we can observe or ever hope to observe. That's exactly what infinity is. Not a number per se, but rather a size. The size of something that doesn't end. Infinity is not the biggest number, instead, it is how many numbers there are. But there are different sizes of infinity. The smallest type of infinity is countable infinity. The number of hours in forever. It's also the number of whole numbers that there are, natural number, the numbers we use when counting things, like 1, 2, 3, 4, 5, 6 and so on. Sets like these are unending, but they are countable. Countable means that you can count them from one element to any other in a finite amount of time, even if that finite amount of time is longer than you will live or the universe will exist for, it's still finite. Uncountable infinity, on the other hand, is literally bigger. Too big to even count. The number of real numbers that there are, not just whole numbers, but all numbers is uncountably infinite. You literally cannot count even from 0 to 1 in a finite amount of time by naming every real number in between. I mean, where do you even start? Zero, okay. But what comes next? 0.000000... Eventually, we would imagine a 1 going somewhere at the end, but there is no end. We could always add another 0. Uncountability makes this set so much larger than the set of all whole numbers that even between 0 and 1, there are more numbers than there are whole numbers on the entire endless number line. Georg Cantor's famous diagonal argument helps illustrate this. Imagine listing every number between zero and one. Since they are uncountable and can't be listed in order, let's imagine randomly generating them forever with no repeats. Each number regenerate can be paired with a whole number. If there's a one to one correspondence between the two, that is if we can match one whole number to each real number on our list, that would mean that countable and uncountable sets are the same size. But we can't do that, even though this list goes on for ever. Forever isn't enough. Watch this. If we go diagonally down our endless list of real numbers and take the first decimal of the first number and the second of the second number, the third of the third and so on and add one to each, subtracting one if it happens to be a nine, we can generate a new real number that is obviously between 0 and 1, but since we've defined it to be different from every number on our endless list and at least one place it's clearly not contained in the list. In other words, we've used up every single whole number, the entire infinity of them and yet we can still come up with more real numbers. Here's something else that is true but counterintuitive. There are the same number of even numbers as there are even and odd numbers. At first, that sounds ridiculous. Clearly, there are only half as many even numbers as all whole numbers, but that intuition is wrong. The set of all whole numbers is denser but every even number can be matched with a whole number. You will never run out of members either set, so this one to one correspondence shows that both sets are the same size. In other words, infinity divided by two is still infinity. Infinity plus one is also infinity. A good illustration of this is Hilbert's paradox up the Grand Hotel. Imagine a hotel with a countably infinite number of rooms. But now, imagine that there is a person booked into every single room. Seemingly, it's fully booked, right? No. Infinite sets go against common sense. You see, if a new guest shows up and wants a room, all the hotel has to do is move the guest in room number 1 to room number 2. And a guest in room 2 to room 3 and 3 to 4 and 4 to 5 and so on. Because the number of rooms is never ending we cannot run out of rooms. Infinity 1 is also infinity again. If one guest leaves the hotel, we can shift every guest the other way. Guest 2 goes to room 1, 3 to 2, 4 to 3 and so on, because we have an infinite amount of guests. That is a never ending supply of them. No room will be left empty. As it turns out, you can subtract any finite number from infinity and still be left with infinity. It doesn't care. It's unending. BanachTarski hasn't left our sights yet. All of this is related. We are now ready to move on to shapes. Hilbert's hotel can be applied to a circle. Points around the circumference can be thought of as guests. If we remove one point from the circle that point is gone, right? Infinity tells us it doesn't matter. The circumference of a circle is irrational. It's the radius times 2Pi. So, if we mark off points beginning from the whole, every radius length along the circumference going clockwise we will never land on the same point twice, ever. We can count off each point we mark with a whole number. So this set is neverending, but countable, just like guests and rooms in Hilbert's hotel. And like those guests, even though one has checked out, we can just shift the rest. Move them counterclockwise and every room will be filled Point 1 moves to fill in the hole, point 2 fills in the place where point 1 used to be, 3 fills in 2 and so on. Since we have a unending supply of numbered points, no hole will be left unfilled. The missing point is forgotten. We apparently never needed it to be complete. There's one last needo consequence of infinity we should discuss before tackling BanachTarski. Ian Stewart famously proposed a brilliant dictionary. One that he called the Hyperwebster. The Hyperwebster lists every single possible word of any length formed from the 26 letters in the English alphabet. It begins with "a," followed by "aa," then "aaa," then "aaaa." And after an infinite number of those, "ab," then "aba," then "abaa", "abaaa," and so on until "z, "za," "zaa," et cetera, et cetera, until the final entry in infinite sequence of "z"s. Such a dictionary would contain every single word. Every single thought, definition, description, truth, lie, name, story. What happened to Amelia Earhart would be in that dictionary, as well as every single thing that didn't happened to Amelia Earhart. Everything that could be said using our alphabet. Obviously, it would be huge, but the company publishing it might realize that they could take a shortcut. If they put all the words that begin with a in a volume titled "A," they wouldn't have to print the initial "a." Readers would know to just add the "a," because it's the "a" volume. By removing the initial "a," the publisher is left with every "a" word sans the first "a," which has surprisingly become every possible word. Just one of the 26 volumes has been decomposed into the entire thing. It is now that we're ready to investigate this video's titular paradox. What if we turned an object, a 3D thing into a Hyperwebster? Could we decompose pieces of it into the whole thing? Yes. The first thing we need to do is give every single point on the surface of the sphere one name and one name only. A good way to do this is to name them after how they can be reached by a given starting point. If we move this starting point across the surface of the sphere in steps that are just the right length, no matter how many times or in what direction we rotate, so long as we never backtrack, it will never wind up in the same place twice. We only need to rotate in four directions to achieve this paradox. Up, down, left and right around two perpendicular axes. We are going to need every single possible sequence that can be made of any finite length out of just these four rotations. That means we will need lef, right, up and down as well as left left, left up, left down, but of course not left right, because, well, that's backtracking. Going left and then right means you're the same as you were before you did anything, so no left rights, no right lefts and no up downs and no down ups. Also notice that I'm writing the rotations in order right to left, so the final rotation is the leftmost letter. That will be important later on. Anyway. A list of all possible sequences of allowed rotations that are finite in lenght is, well, huge. Countably infinite, in fact. But if we apply each one of them to a starting point in green here and then name the point we land on after the sequence that brought us there, we can name a countably infinite set of points on the surface. Let's look at how, say, these four strings on our list would work. Right up left. Okay, rotating the starting point this way takes us here. Let's colour code the point based on the final rotation in its string, in this case it's left and for that we will use purple. Next up down down. That sequence takes us here. We name the point DD and color it blue, since we ended with a down rotation. RDR, that will be this point's name, takes us here. And for a final right rotation, let's use red. Finally, for a sequence that end with up, let's colour code the point orange. Now, if we imagine completing this process for every single sequence, we will have a countably infinite number of points named and colorcoded. That's great, but not enough. There are an uncountably infinite number of points on a sphere's surface. But no worries, we can just pick a point we missed. Any point and color it green, making it a new starting point and then run every sequence from here. After doing this to an uncountably infinite number of starting point we will have indeed named and colored every single point on the surface just once. With the exception of poles. Every sequence has two poles of rotation. Locations on the sphere that come back to exactly where they started. For any sequence of right or left rotations, the polls are the north and south poles. The problem with poles like these is that more than one sequence can lead us to them. They can be named more than once and be colored in more than one color. For example, if you follow some other sequence to the north or south pole, any subsequent rights or lefts will be equally valid names. In order to deal with this we're going to just count them out of the normal scheme and color them all yellow. Every sequence has two, so there are a countably infinite amount of them. Now, with every point on the sphere given just one name and just one of six colors, we are ready to take the entire sphere apart. Every point on the surface corresponds to a unique line of points below it all the way to the center point. And we will be dragging every point's line along with it. The lone center point we will set aside. Okay, first we cut out and extract all the yellow poles, the green starting points, the orange up points, the blue down points and the red and purple left and right points. That's the entire sphere. With just these pieces you could build the whole thing. But take a look at the left piece. It is defined by being a piece composed of every point, accessed via a sequence ending with a left rotation. If we rotate this piece right, that's the same as adding an "R" to every point's name. But left and then right is a backtrack, they cancel each other out. And look what happens when you reduce them away. The set becomes the same as a set of all points with names that end with L, but also U, D and every point reached with no rotation. That's the full set of starting points. We have turned less than a quarter of the sphere into nearly threequarters just by rotating it. We added nothing. It's like the Hyperwebster. If we had the right piece and the poles of rotation and the center point, well, we've got the entire sphere again, but with stuff left over. To make a second copy, let's rotate the up piece down. The down ups cancel because, well, it's the same as going nowhere and we're left with a set of all starting points, the entire up piece, the right piece and the left piece, but there's a problem here. We don't need this extra set of starting points. We still haven't used the original ones. No worries, let's just start over. We can just move everything from the up piece that turns into a starting point when rotated down. That means every point whose final rotation is up. Let's put them in the piece. Of course, after rotating points named UU will just turn into points named U, and that would give us a copy here and here. So, as it turns out, we need to move all points with any name that is just a string of Us. We will put them in the down piece and rotate the up piece down, which makes it congruent to the up right and left pieces, add in the down piece along with some up and the starting point piece and, well, we're almost done. The poles of rotation and center are missing from this copy, but no worries. There's a countably infinite number of holes, where the poles of rotations used to be, which means there is some pole around which we can rotate this sphere such that every pole hole orbits around without hitting another. Well, this is just a bunch of circles with one point missing. We fill them each like we did earlier. And we do the same for the centerpoint. Imagine a circle that contains it inside the sphere and just fill in from infinity and look what we've done. We have taken one sphere and turned it into two identical spheres without adding anything. One plus one equals 1. That took a while to go through, but the implications are huge. And mathematicians, scientists and philosophers are still debating them. Could such a process happen in the real world? I mean, it can happen mathematically and math allows us to abstractly predict and describe a lot of things in the real world with amazing accuracy, but does the BanachTarski paradox take it too far? Is it a place where math and physics separate? We still don't know. History is full of examples of mathematical concepts developed in the abstract that we did not think would ever apply to the real world for years, decades, centuries, until eventually science caught up and realized they were totally applicable and useful. The BanachTarski paradox could actually happen in our realworld, the only catch of course is that the five pieces you cut your object into aren't simple shapes. They must be infinitely complex and detailed. That's not possible to do in the real world, where measurements can only get so small and there's only a finite amount of time to do anything, but math says it's theoretically valid and some scientists think it may be physically valid too. There have been a number of papers published suggesting a link between by BanachTarski and the way tiny tiny subatomic particles can collide at high energies and turn into more particles than we began with. We are finite creatures. Our lives are small and can only scientifically consider a small part of reality. What's common for us is just a sliver of what's available. We can only see so much of the electromagnetic spectrum. We can only delve so deep into extensions of space. Common sense applies to that which we can access. But common sense is just that. Common. If total sense is what we want, we should be prepared to accept that we shouldn't call infinity weird or strange. The results we've arrived at by accepting it are valid, true within the system we use to understand, measure, predict and order the universe. Perhaps the system still needs perfecting, but at the end of day, history continues to show us that the universe isn't strange. We are. And as always, thanks for watching. Finally, as always, the description is full of links to learn more. There are also a number of books linked down there that really helped me wrap my mind kinda around BanachTarski. First of all, Leonard Wapner's "The Pea and the Sun." This book is fantastic and it's full of lot of the preliminaries needed to understand the proof that comes later. He also talks a lot about the ramifications of what BanachTarski and their theorem might mean for mathematics. Also, if you wanna talk about math and whether it's discovered or invented, whether it really truly will map onto the universe, Yanofsky's "The Outer Limits of Reason" is great. This is the favorite book of mine that I've read this entire year. Another good one is E. Brian Davies' "Why Beliefs Matter." This is actually Corn's favorite book, as you might be able to see there. It's delicious and full of lots of great information about the limits of what we can know and what science is and what mathematics is. If you love infinity and math, I cannot more highly recommend Matt Parker's "Things to Make and Do in the Fourth Dimension." He's hilarious and this book is very very great at explaining some pretty awesome things. So keep reading, and if you're looking for something to watch, I hope you've already watched Kevin Lieber's film on Field Day. I already did a documentary about Whittier, Alaska over there. Kevin's got a great short film about putting things out on the Internet and having people react to them. There's a rumor that Jake Roper might be doing something on Field Day soon. So check out mine, check out Kevin's and subscribe to Field Day for upcoming Jake Roper action, yeah? He's actually in this room right now, say hi, Jake. [Jake:] Hi. Thanks for filming this, by the way. Guys, I really appreciate who you all are. And as always, thanks for watching.
Contents
Definition
A topological space $X$ is called countably generated if for every subset $V\subseteq X,$ $V$ is closed in $X$ whenever for each countable subspace $U$ of $X$ the set $V\cap U$ is closed in $U$. Equivalently, $X$ is countably generated if and only if the closure of any $A\subseteq X$ equals the union of closures of all countable subsets of $A.$
Countable fan tightness
A topological space $X$ has countable fan tightness if for every point $x\in X$ and every sequence $A_{1},A_{2},\ldots$ of subsets of the space $X$ such that $x\in \bigcap _{n}{\overline {A_{n}}},$ there are finite set $B_{1}\subseteq A_{1},B_{2}\subseteq A_{2},\ldots$ such that $x\in {\overline {\bigcup _{n}B_{n}}}.$
A topological space $X$ has countable strong fan tightness if for every point $x\in X$ and every sequence $A_{1},A_{2},\ldots$ of subsets of the space $X$ such that $x\in \bigcap _{n}{\overline {A_{n}}},$ there are points $x_{1}\subseteq A_{1},x_{2}\subseteq A_{2},\ldots$ such that $x\in {\overline {\left\{x_{1},x_{2},\ldots \right\}}}.$ Every strong Fréchet–Urysohn space has strong countable fan tightness.
Properties
A quotient of a countably generated space is again countably generated. Similarly, a topological sum of countably generated spaces is countably generated. Therefore the countably generated spaces form a coreflective subcategory of the category of topological spaces. They are the coreflective hull of all countable spaces.
Any subspace of a countably generated space is again countably generated.
Examples
Every sequential space (in particular, every metrizable space) is countably generated.
An example of a space which is countably generated but not sequential can be obtained, for instance, as a subspace of Arens–Fort space.
See also
 Finitely generated space
 Tightness (topology) − Tightness is a cardinal function related to countably generated spaces and their generalizations.
References
 Herrlich, Horst (1968). Topologische Reflexionen und Coreflexionen. Lecture Notes in Math. 78. Berlin: Springer.
External links
 A Glossary of Definitions from General Topology [1]
 https://web.archive.org/web/20040917084107/http://thales.doa.fmph.uniba.sk/density/pages/slides/sleziak/paper.pdf
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