I lied
Okay, so I know I said that this post would be about the criticism and pushback I got from my work, but I just couldn't help myself from sharing another really cool thing I thought of!! So, yes, I lied, this blog post will not be about what I said it would, however, I expect to make well on that statement for the one right after this.
With that being said, let's get into it.
When you think of infinity, what do you think about? I asked one of my friends this and he mentioned "that sort of continuous x-axis line/arrow that just keeps going and going and going and never stops". I imagine this is a fairly common initial perception of infinity, which is all fine and dandy because we won't really be talking about infinities this time (well, we sort of will, but it isn't the main point). So to start, I'll explain to you what a perfect set is. A perfect set is a set in space which has no isolated points and is closed. This basically means that if you look at any point within that set, you can find sets of points that get closer and closer (converge) to that point no matter how small of a neighborhood you look at (no matter how much you "zoom in" on that point). So if no point is isolated, then there are an infinite amount of points (which are also in the set) that are near/converge to that point. This is like if you draw a line with you pencil and zoom in on any segment of it, you'll always see graphite. Then that closed part basically just means that the points on it's ends/boundaries are also contained within the set.
Okay, so now that you have an idea of what classifies a perfect set, try to imagine one yourself. Maybe you are still thinking of that line you drew with your pencil, so let's just go with that, and maybe simplify it to a more concrete thing: think of the line that covers the closed interval [0,1]. So we're looking at drawing a line that starts at 0 and ends at 1, including both 0 and 1. Well it's pretty obvious to see that if you pick any 2 spots on that line, you can always find another spot in between those two: if you look at some point on the left side, say maybe .2, and another point on the right side, say maybe .7, you can see there's a wide range of things in between them, like .3, .4, .6, .401, etc. This will work for this interval no matter how close you pick those 2 spots on the line (for example .49999998 and .49999999, you can find the point .499999985 that lies between them).
This is what it means for a set to be dense. For any x,y in your set (say, without loss of generality that x<y), you can find a point z also in your set such that x<z<y. This is what we just showed, that the interval [0,1] is dense. And this makes perfect sense! The set of all points within [0,1] is a perfect set, so if no points are isolated, how could you not find a point in between any other points?
Well, let me introduce you to my Cantor set. A set which is perfect, yet not dense. Moreover, not only is it not dense, but it is NOWHERE dense. Let me walk you through how we create such a set, starting with our familiar interval [0,1]. Well, if we are looking for a set that's nowhere dense, then we can't have any point between two other points. In other words, we need some gaps, right? Alright, so let's make a gap: take out/erase the middle third of that interval. So if we have [0,1]\(1/3,2/3), we get 2 parts: [0,1/3] U [2/3,1]. But Georg, what about the points within those intervals like 1/6, or 5/6? Alright then, let's just cut those out too, remove the middle third of both [0,1/3] and [2/3,1]. Now we have the set {[0,1/9] U [2/9,1/3] U [2/3,7/9] U [8/9,1]}. Now continue doing this process of removing the middle third over and over and over again ad infinitum. Here's a visualization of the first 6 iterations:
The points remaining in these black intervals are what is left when we keep deleting the middle third. The set we construct will be the intersection of all of these black interval's unions (so consider our set to be where each of the black intervals shown intersect/share an interval with any of the intervals in any of the rows/iterations above/below it.
Because of how we have defined this set, each point within the set is a limit point of the set (by being the intersection of a neighborhood (albeit a deleted neighborhood possibly) around that point and the set itself is non-empty (just look above the deleted neighborhood)). Since each point is a limit point, and we have defined it to be closed, our set is perfect. Yet, since we are repeatedly deleting many points in between other points infinitely (so I suppose I lied again, since infinity does have a pretty big part to play), it is also nowhere dense.
With that being said, let's get into it.
When you think of infinity, what do you think about? I asked one of my friends this and he mentioned "that sort of continuous x-axis line/arrow that just keeps going and going and going and never stops". I imagine this is a fairly common initial perception of infinity, which is all fine and dandy because we won't really be talking about infinities this time (well, we sort of will, but it isn't the main point). So to start, I'll explain to you what a perfect set is. A perfect set is a set in space which has no isolated points and is closed. This basically means that if you look at any point within that set, you can find sets of points that get closer and closer (converge) to that point no matter how small of a neighborhood you look at (no matter how much you "zoom in" on that point). So if no point is isolated, then there are an infinite amount of points (which are also in the set) that are near/converge to that point. This is like if you draw a line with you pencil and zoom in on any segment of it, you'll always see graphite. Then that closed part basically just means that the points on it's ends/boundaries are also contained within the set.
Okay, so now that you have an idea of what classifies a perfect set, try to imagine one yourself. Maybe you are still thinking of that line you drew with your pencil, so let's just go with that, and maybe simplify it to a more concrete thing: think of the line that covers the closed interval [0,1]. So we're looking at drawing a line that starts at 0 and ends at 1, including both 0 and 1. Well it's pretty obvious to see that if you pick any 2 spots on that line, you can always find another spot in between those two: if you look at some point on the left side, say maybe .2, and another point on the right side, say maybe .7, you can see there's a wide range of things in between them, like .3, .4, .6, .401, etc. This will work for this interval no matter how close you pick those 2 spots on the line (for example .49999998 and .49999999, you can find the point .499999985 that lies between them).
This is what it means for a set to be dense. For any x,y in your set (say, without loss of generality that x<y), you can find a point z also in your set such that x<z<y. This is what we just showed, that the interval [0,1] is dense. And this makes perfect sense! The set of all points within [0,1] is a perfect set, so if no points are isolated, how could you not find a point in between any other points?
Well, let me introduce you to my Cantor set. A set which is perfect, yet not dense. Moreover, not only is it not dense, but it is NOWHERE dense. Let me walk you through how we create such a set, starting with our familiar interval [0,1]. Well, if we are looking for a set that's nowhere dense, then we can't have any point between two other points. In other words, we need some gaps, right? Alright, so let's make a gap: take out/erase the middle third of that interval. So if we have [0,1]\(1/3,2/3), we get 2 parts: [0,1/3] U [2/3,1]. But Georg, what about the points within those intervals like 1/6, or 5/6? Alright then, let's just cut those out too, remove the middle third of both [0,1/3] and [2/3,1]. Now we have the set {[0,1/9] U [2/9,1/3] U [2/3,7/9] U [8/9,1]}. Now continue doing this process of removing the middle third over and over and over again ad infinitum. Here's a visualization of the first 6 iterations:
The points remaining in these black intervals are what is left when we keep deleting the middle third. The set we construct will be the intersection of all of these black interval's unions (so consider our set to be where each of the black intervals shown intersect/share an interval with any of the intervals in any of the rows/iterations above/below it.
Because of how we have defined this set, each point within the set is a limit point of the set (by being the intersection of a neighborhood (albeit a deleted neighborhood possibly) around that point and the set itself is non-empty (just look above the deleted neighborhood)). Since each point is a limit point, and we have defined it to be closed, our set is perfect. Yet, since we are repeatedly deleting many points in between other points infinitely (so I suppose I lied again, since infinity does have a pretty big part to play), it is also nowhere dense.
Links for image sources: https://en.wikipedia.org/wiki/Cantor_set
ReplyDeletehttps://www.missouriwestern.edu/orgs/momaa/ChrisShaver-CantorSetPaper4.pdf
Link for some info: https://arxiv.org/pdf/1411.7110.pdf