Information about Manifolds - Part 1

00:00 Introduction

00:20 Overview

02:24 Stoke’s theorem as the goal

02:56 Metric Spaces

04:56 Definition Topology

07:29 Simple examples of topological spaces

09:07 Credits

1 00:00:00,800 –> 00:00:07,040 Hello and welcome to Manifolds, a video series  I started for everyone who wants to learn how  

2 00:00:07,040 –> 00:00:14,240 to use calculus on surfaces and related topics.  However, before we start I really want to thank  

3 00:00:14,240 –> 00:00:20,160 all the nice people who support this channel  on Steady, via Paypal, or by other means.  

4 00:00:20,160 –> 00:00:25,760 Now, in part 1 of the course, I give you a short  overview and explain the definition of a topology. 

5 00:00:26,640 –> 00:00:33,280 So let’s start with a quick explanation what we  will discuss in this course. First, as I’ve already  

6 00:00:33,280 –> 00:00:40,880 mentioned, the notion of a manifold generalizes the  concept of a surface in space. Such a surface we  

7 00:00:40,880 –> 00:00:47,360 could visualize like this but the most popular  one would be the two-dimensional sphere.  

8 00:00:47,360 –> 00:00:54,880 So this is the boundary of the ball and often called  S^2. The 2 stands for the two dimensions we have  

9 00:00:54,880 –> 00:01:01,040 when we just live on the surface. Now you might  already know some important applications where  

10 00:01:01,040 –> 00:01:08,800 it is needed to calculate on the sphere. Especially in physics, it happens that some constraints force  

11 00:01:08,800 –> 00:01:16,400 the motion to happen on the surface. Then questions  like finding a minimum or a maximum of a function  

12 00:01:16,400 –> 00:01:21,520 are completely different because we can’t  use our calculus for open domains anymore.

13 00:01:22,400 –> 00:01:28,240 Therefore the overall question here will  be how to extend our calculation rules for  

14 00:01:28,240 –> 00:01:35,200 surfaces and indeed abstract manifolds. In order  to do this we first have to understand what the  

15 00:01:35,200 –> 00:01:42,160 fundamentals of such a surface are. So we start  with a quick overview of the field of topology.

16 00:01:43,200 –> 00:01:48,240 Afterwards we will be able to define  differentiable manifolds as our subject of  

17 00:01:48,240 –> 00:01:56,240 study. So you see the notion of differentiability  will be a crucial point in this course. Then next  

18 00:01:56,240 –> 00:02:03,280 for these differentiable manifolds we will be able  to define so-called differential forms. At first  

19 00:02:03,280 –> 00:02:09,360 glance these differential forms might look strange  because they are just given by one part of an  

20 00:02:09,360 –> 00:02:18,080 integral like dx or d omega. However we will  define these objects in a rigorous way and indeed,  

21 00:02:18,080 –> 00:02:26,000 in the end, also integrals with these differential  forms will make sense. Okay now the overall goal  

22 00:02:26,000 –> 00:02:32,800 I have in mind for this course here is that we  will reach the generalized Stokes’s theorem. Indeed  

23 00:02:32,800 –> 00:02:39,200 this theorem will nicely form a connection between  a manifold and its boundary by using differential  

24 00:02:39,200 –> 00:02:46,160 form. So this is the overview of the course and  i think we are ready to start with the first part  

25 00:02:46,160 –> 00:02:52,960 and talk about topology. Now if you already have  a good knowledge of metric spaces you can use  

26 00:02:52,960 –> 00:02:59,840 this as a starting point. Indeed a lot of notions  we have in topology are already formulated  

27 00:02:59,840 –> 00:03:07,040 in metric spaces. Here please recall: a metric  space needs a set X and a distance function d.  

28 00:03:08,000 –> 00:03:13,360 This means when we have to set X we can  measure distances between two points in X  

29 00:03:14,320 –> 00:03:19,280 for example here x and y have a  distance given by a positive real number  

30 00:03:20,080 –> 00:03:27,040 and this one is denoted by d(x,y). Now in the  case you see this the first time you can watch  

31 00:03:27,040 –> 00:03:32,320 the first videos in my functional analysis  course to get familiar with metric spaces  

32 00:03:33,280 –> 00:03:40,000 however the important thing here is that you  know how we can define open sets usually this  

33 00:03:40,000 –> 00:03:47,760 works when we take so called open epsilon  balls. This means that B_epsilon(x) is a ball  

34 00:03:47,760 –> 00:03:55,760 with radius epsilon and middle point x. Now by  using these epsilon balls we can say if a subset  

35 00:03:55,760 –> 00:04:04,320 in the metric space is open. Hence here the notion  “open” for a subset depends on the chosen metric d. 

36 00:04:05,360 –> 00:04:11,840 But then we can show that the collection of  all open sets fulfills some nice properties. 

37 00:04:11,840 –> 00:04:19,280 For example if we take two open sets the intersection  is always also an open set with such properties  

38 00:04:19,280 –> 00:04:24,960 in mind we see that for a lot of things we  do not need an explicit measure of distance  

39 00:04:25,760 –> 00:04:32,400 just some neighborhood relation between the points  might be sufficient so roughly speaking we just  

40 00:04:32,400 –> 00:04:38,800 need to know which points are neighbors of x are  close to x without measuring the explicit distance  

41 00:04:40,000 –> 00:04:43,920 indeed the abstraction of  this idea leads to topology  

42 00:04:44,880 –> 00:04:50,640 hence we just list all the sets that should be  open and then we deduce everything from them  

43 00:04:51,920 –> 00:04:55,840 most importantly in this definition  we don’t need a metric anymore  

44 00:04:57,200 –> 00:05:04,720 however of course we still have a set X then what  we need is the collection of all subsets of X 

45 00:05:04,720 –> 00:05:11,920 which we call the power set of X and denote  by P(X) therefore to say which sets are open  

46 00:05:11,920 –> 00:05:18,240 we just have to take a subset of the power  set and this one is denoted by a curved T

47 00:05:19,360 –> 00:05:26,720 in other words this T just stands for a collection  of subsets of X now these subsets should be the  

48 00:05:26,720 –> 00:05:33,680 open sets therefore they have to fulfill all the  rules like in the metric space indeed we will  

49 00:05:33,680 –> 00:05:40,400 fix three important properties here now the first  one is very simple we just say that the empty set  

50 00:05:40,400 –> 00:05:47,120 and the whole space X are open sets. More  precisely they are elements of the collection T

51 00:05:48,080 –> 00:05:53,360 then the second property I already  mentioned. If we take two open sets A and B

52 00:05:54,960 –> 00:06:02,480 then this implies that the intersection is also  an open set finally the third and last property  

53 00:06:02,480 –> 00:06:09,600 looks similarly but now for the union however you  might know in a metric space with the union and  

54 00:06:09,600 –> 00:06:16,320 open sets we can do a lot. What I mean is it does  not matter how many open sets are in the union  

55 00:06:17,040 –> 00:06:24,080 the result is always an open set as well hence  here we can look at the whole family of open sets  

56 00:06:25,280 –> 00:06:31,680 so we look at A_i where i goes through any  fixed index set capital I and of course  

57 00:06:31,680 –> 00:06:38,160 any subset in the family is an element of  t and then this implies that we can look at  

58 00:06:38,160 –> 00:06:45,920 the big union i in i of the sets A_i and we  conclude this union is also an element in T

59 00:06:46,880 –> 00:06:52,400 okay and there you see we condense the  properties of open sets for a metric space

60 00:06:52,400 –> 00:06:58,960 into a new definition. Indeed with this definition  we can work and we don’t need a metric anymore  

61 00:06:59,760 –> 00:07:06,720 what we now have is a collection of subsets T  and we call it a topology on X and I already  

62 00:07:06,720 –> 00:07:14,400 mentioned it a lot the elements of a topology are  called open sets therefore please always remember  

63 00:07:14,400 –> 00:07:21,600 in the topology the property open is given by  definition and therefore open only makes sense  

64 00:07:21,600 –> 00:07:28,720 with respect to a chosen topology. OK, I think it  will be very helpful when we look at some examples  

65 00:07:29,600 –> 00:07:36,720 therefore let’s start with the easiest examples so  the question is what is the simplest choice for T  

66 00:07:36,720 –> 00:07:42,160 such that all the rules are fulfilled. Of course in order to satisfy the first rule  

67 00:07:42,160 –> 00:07:48,880 we need at least the empty set and X involved  however if we leave it at that we already have  

68 00:07:48,880 –> 00:07:55,680 a topology so what you should see is the second  and the third property are immediately satisfied  

69 00:07:56,480 –> 00:08:00,240 simply because there are not many choices  for the intersection and the union  

70 00:08:01,040 –> 00:08:07,040 hence this means that this is the topology  where all the non-trivial subsets are not open  

71 00:08:08,000 –> 00:08:14,400 so maybe not the most interesting topology  to work with ok now you might already know  

72 00:08:14,400 –> 00:08:21,200 we can also do the other extreme which means we  have the topology where all the subsets are open  

73 00:08:22,160 –> 00:08:28,240 of course the power set of X is a topology  on X because there’s no way to violate one  

74 00:08:28,240 –> 00:08:35,680 of these rules the power set just contains  all the subsets and now in this topology all  

75 00:08:35,680 –> 00:08:40,400 these subsets are open sets, therefore  we often call it the discrete topology.

76 00:08:41,360 –> 00:08:48,000 On the other hand the first example is often  called the indiscrete topology. Of course both  

77 00:08:48,000 –> 00:08:53,120 examples are not the most interesting topologies  but they are edge cases you should always have  

78 00:08:53,120 –> 00:08:58,800 in mind. Okay maybe that’s good enough for the  introduction today. Let’s continue in the next  

79 00:08:58,800 –> 00:09:07,840 video while working with the open sets. Therefore  I hope I see you there and have a nice day. Bye!

80 00:09:20,560 –> 00:09:21,060

Q1: Let $X$ be a set and $\mathcal{T}$ be a topology on $X$. Which claim is correct?

A1: $\mathcal{T} \in \mathcal{P}(X)$

A2: $\mathcal{T} \in X$

A3: $\mathcal{T} \subseteq \mathcal{P}(X)$

A4: $\mathcal{T} \subseteq X$

A5: $\mathcal{T} = X$

Q2: Let $X$ be a set and $\mathcal{T}$ be a topology on $X$. Which claim is not correct?

A1: $\emptyset \in \mathcal{T}$

A2: $X \in \mathcal{T}$

A3: ${\emptyset, X} \subseteq \mathcal{T}$

A4: $\mathcal{P}(X) \in \mathcal{T}$

Q3: Let $X$ be a set and $\mathcal{T}$ be a topology on $X$. Which claim is, in general, not correct?

A1: If $A_j \in \mathcal{T} $ for $j \in \mathbb{N}$, then $\bigcap_{j \in \mathbb{N} } A_j \in \mathcal{T}$.

A2: If $A_j \in \mathcal{T} $ for $j \in \mathbb{N}$, then $\bigcup_{j \in \mathbb{N} } A_j \in \mathcal{T}$.

A3: If $A_j \in \mathcal{T} $ for $j =1,2,3$, then $\bigcap_{j=1}^3 A_j \in \mathcal{T}$.