-
Title: Open, Closed and Compact Sets
-
Series: Real Analysis
-
Chapter: Sequences and Limits
-
YouTube-Title: Real Analysis 13 | Open, Closed and Compact Sets
-
Bright video: https://youtu.be/Wqo4Svs4erw
-
Dark video: https://youtu.be/5huKNKrZjYM
-
Ad-free video: Watch Vimeo video
-
Forum: Ask a question in Mattermost
-
Quiz: Test your knowledge
-
Dark-PDF: Download PDF version of the dark video
-
Print-PDF: Download printable PDF version
-
Exercise Download PDF sheets
-
Thumbnail (bright): Download PNG
-
Thumbnail (dark): Download PNG
-
Subtitle on GitHub: ra13_sub_eng.srt
-
Download bright video: Link on Vimeo
-
Download dark video: Link on Vimeo
-
Timestamps
00:00 Intro
00:14 Recalling (epsilon-)neighbourhoods
01:33 Example: neighbourhoods
02:43 Definition open sets
04:00 Definition closed set
04:43 Examples
06:11 Criterion for checking closeness with the help of sequences
06:44 Example for the criterion
07:40 Definition compact sets
08:58 Credits
-
Subtitle in English
1 00:00:00,429 –> 00:00:03,315 Hello and welcome back to real analysis.
2 00:00:04,043 –> 00:00:08,876 Of course as always I want to thank all the nice people that support this channel on Steady or Paypal.
3 00:00:09,429 –> 00:00:14,613 In today’s part 13 we will talk about open, closed and compact sets.
4 00:00:14,813 –> 00:00:20,925 First please recall that for any point x on the number line, we have the epsilon-neighbourhoods.
5 00:00:21,214 –> 00:00:24,914 They are defined for positive epsilons as intervals.
6 00:00:25,686 –> 00:00:30,519 Namely we start at (x - epsilon) and go to (x + epsilon)
7 00:00:30,719 –> 00:00:34,348 and then they are simply called the epsilon-neighbourhood of x
8 00:00:34,986 –> 00:00:39,746 and a common notation one uses is B with index epsilon of x.
9 00:00:39,946 –> 00:00:47,329 The important thing you should remember here is that all the points that are close to x, are inside this B_epsilon(x)
10 00:00:48,086 –> 00:00:52,170 and this closeness is just quantified with this value epsilon.
11 00:00:52,586 –> 00:00:56,357 Which gives us the maximum distance the points can have from x.
12 00:00:57,000 –> 00:01:02,855 However if you don’t want or need to quantify the closeness, there is another notion one uses.
13 00:01:03,414 –> 00:01:06,609 Namely we simply call it a neighbourhood of x
14 00:01:07,214 –> 00:01:13,848 and this could be any subset of the real numbers, as long as we find the B_epsilon(x) inside
15 00:01:14,048 –> 00:01:23,736 or in other words: we need to find a positive number epsilon, such that the epsilon-neighbourhood of x, B_epsilon(x), is a subset of M.
16 00:01:23,936 –> 00:01:32,616 So you see the notion of neighbourhood of x is very general, but the crucial thing is that we find a normal epsilon-neighbourhood of x inside of it.
17 00:01:32,943 –> 00:01:35,986 Ok, maybe we should start with a simple example.
18 00:01:36,757 –> 00:01:41,008 If you have the number line in mind, we immediately get a lot of subsets.
19 00:01:41,208 –> 00:01:46,242 For example the interval that starts with the number -2 and goes to the number 2.
20 00:01:47,200 –> 00:01:51,222 This set is the neighbourhood of the point x = 0.
21 00:01:51,700 –> 00:01:56,309 However, also it’s a neighbourhood of the point x = 1.
22 00:01:56,509 –> 00:02:02,389 The only important thing here is that we find an epsilon, that does not matter how large it is.
23 00:02:02,914 –> 00:02:05,200 It only needs to be positive.
24 00:02:05,686 –> 00:02:10,765 So maybe here in the second case we have to choose a smaller epsilon, than before.
25 00:02:10,965 –> 00:02:14,948 However still, the only thing we need is that we find such an epsilon
26 00:02:15,771 –> 00:02:20,937 and we do not find such a number, if the point x is given by 2.
27 00:02:21,137 –> 00:02:28,096 Of course 2 is an element of this set, but an epsilon-neighbourhood around it lies not in the set.
28 00:02:28,296 –> 00:02:34,811 No matter how small the epsilon is, we always find a part of the epsilon-neighbourhood that lies outside.
29 00:02:35,011 –> 00:02:43,170 In summary, here we have an example of a set that is a neighbourhood of some elements of it, but for others it’s not
30 00:02:43,370 –> 00:02:49,140 and now a nice set, that is for all its members a neighbourhood, we call an open set.
31 00:02:49,814 –> 00:02:53,101 So these nice sets will get a special name.
32 00:02:53,301 –> 00:02:58,376 Later we will see, it’s much better to work with open sets, than just with sets.
33 00:02:58,986 –> 00:03:05,392 Ok, now any subset of the real numbers is called open or more precisely open in R,
34 00:03:05,592 –> 00:03:10,203 if for each point x in M, M is a neighbourhood of this point.
35 00:03:11,114 –> 00:03:15,982 Hence such boundary points like this one, are not in the set M itself.
36 00:03:16,182 –> 00:03:21,958 Equivalently for the definition you could avoid the notion neighbourhood and just use the epsilon-neighbourhoods.
37 00:03:22,386 –> 00:03:30,423 So this means for all x in M, there exists an epsilon greater than 0, such that B_epsilon(x) is a subset of M.
38 00:03:30,623 –> 00:03:33,870 Of course we can also visualize that on the number line.
39 00:03:34,429 –> 00:03:38,466 So for example here, these 4 parts could be our set M
40 00:03:38,666 –> 00:03:47,670 and then you can just pick any x from this set and then in the case that M is open, you find a small interval around this point
41 00:03:47,870 –> 00:03:50,073 that is completely inside the set M.
42 00:03:50,273 –> 00:03:56,385 Hence in this picture, these boundary points here can’t be a part of the set M.
43 00:03:56,585 –> 00:03:59,258 Otherwise it would not be an open set.
44 00:03:59,771 –> 00:04:06,453 Now on the other hand a set that contains all these possible boundary points gets also a special name.
45 00:04:07,086 –> 00:04:12,121 Such a set A we call closed or more concretely closed in R.
46 00:04:12,657 –> 00:04:17,808 The definition just reads that the complement of A, A^c is an open set.
47 00:04:18,229 –> 00:04:24,951 For example the interval from before as a closed set, because the complement outside is an open set.
48 00:04:25,471 –> 00:04:31,022 Now one important thing you should really note here is that open is not the opposite of closed.
49 00:04:31,400 –> 00:04:34,683 For example a set could be neither open nor closed
50 00:04:35,200 –> 00:04:39,191 or the other way around, it could be open and closed at the same time.
51 00:04:40,043 –> 00:04:42,954 Ok, maybe that’s a good time to look at some examples.
52 00:04:43,871 –> 00:04:46,946 Let’s start with the simplest subsets you can imagine.
53 00:04:47,714 –> 00:04:50,608 Namely the empty set and R itself.
54 00:04:50,808 –> 00:04:57,077 Of course they are both open, because the condition we have here for openness is immediately fulfilled.
55 00:04:57,277 –> 00:05:03,049 For example for the empty set we don’t have any problem, because there are no elements in the set at all
56 00:05:03,249 –> 00:05:10,401 and on the other hand for the real numbers we don’t have any problem, because all intervals are subsets of the real number line.
57 00:05:10,957 –> 00:05:14,502 However now we also know, they are both closed.
58 00:05:14,929 –> 00:05:20,308 Simply, because the empty set is the complement of the real numbers and vice versa.
59 00:05:21,000 –> 00:05:27,125 Ok, the next example we have already discussed. An interval of this form is closed, but not open.
60 00:05:27,543 –> 00:05:30,800 Therefore we often call such intervals, closed intervals.
61 00:05:31,614 –> 00:05:35,108 Of course then the next example would be an open interval.
62 00:05:35,814 –> 00:05:40,270 So that’s good to know. When we use parentheses here we get an open set.
63 00:05:40,470 –> 00:05:44,551 However in this case it’s not a closed set, which you can prove.
64 00:05:45,043 –> 00:05:48,855 Ok and in the last example here I want to mix the brackets
65 00:05:49,055 –> 00:05:52,338 and in this case we don’t have any of the 2 properties.
66 00:05:53,000 –> 00:05:55,014 It’s neither open nor closed.
67 00:05:55,729 –> 00:05:59,868 Ok, so you see these 2 definitions here are not so complicated at all.
68 00:06:00,271 –> 00:06:06,569 But please keep in mind, a subset of the real numbers could be much more complicated than just an interval.
69 00:06:07,271 –> 00:06:10,904 In order to deal with such sets the next fact is very helpful.
70 00:06:11,243 –> 00:06:15,868 It gives us a criterion to check closeness with the help of sequences.
71 00:06:16,500 –> 00:06:29,477 Namely a set A is closed if and only if for all convergent sequences a_n with the property that all the sequence members lie inside the set A,
72 00:06:29,914 –> 00:06:33,842 we have that the limit lies also in A.
73 00:06:34,500 –> 00:06:40,617 To put this in other words it’s not possible to leave the set with sequences from the inside.
74 00:06:40,817 –> 00:06:44,018 Also here it might be helpful to look at an example.
75 00:06:44,386 –> 00:06:49,977 Here on the the number line let’s take the interval that starts with 0 and ends with 2.
76 00:06:50,986 –> 00:06:54,973 Here the number 2 lies inside the interval, but 0 does not.
77 00:06:55,173 –> 00:06:59,859 Hence we are not able to leave the interval from the inside to the right.
78 00:07:00,059 –> 00:07:05,530 For example we could take such a sequence, which is convergent, but then it would have the limit 2.
79 00:07:05,730 –> 00:07:08,393 It’s not possible to get the limit outside.
80 00:07:08,593 –> 00:07:11,980 However on the left-hand side it’s indeed possible.
81 00:07:12,571 –> 00:07:15,870 For example we could take the sequence 1 over n.
82 00:07:16,586 –> 00:07:20,837 It’s a convergent sequence, where all the sequence members lie inside A.
83 00:07:21,614 –> 00:07:26,528 However, the limit here is 0. Which is not an element of A.
84 00:07:26,728 –> 00:07:30,383 Hence the conclusion is: this set is not closed.
85 00:07:31,157 –> 00:07:35,288 Ok, now you should know the definition of open sets and closed sets.
86 00:07:35,986 –> 00:07:39,985 Therefore you are ready for the next definition about compact sets.
87 00:07:40,671 –> 00:07:47,277 It’s a little bit more complicated, but because you already know sequences, we can use them to describe the definition.
88 00:07:47,477 –> 00:07:51,571 So we call a subset of the real numbers A, compact.
89 00:07:52,243 –> 00:07:57,872 If for all sequences a_n, again all the members should lie inside a set A,
90 00:07:58,629 –> 00:08:02,896 we find a convergent subsequence, where the limit lies in A.
91 00:08:03,096 –> 00:08:07,718 So you should see, this is different to the property of closeness from before.
92 00:08:07,918 –> 00:08:13,015 Indeed here you could say a compact set enforces clustering inside it.
93 00:08:13,514 –> 00:08:20,300 Each sequence inside the set needs to have a cluster point, an accumulation value inside the set.
94 00:08:21,100 –> 00:08:29,478 Now, if you compare this definition to the property of a closed set from before, with the sequences you see, this here requires much more.
95 00:08:30,314 –> 00:08:34,886 Hence you can already keep in mind, a compact set is necessarily closed.
96 00:08:35,386 –> 00:08:37,581 However not the other way around.
97 00:08:38,029 –> 00:08:44,191 Maybe you can immediately find a set that is closed by our definition, but not compact with this definition.
98 00:08:44,514 –> 00:08:50,778 Ok, then in the next video I will show you how we can describe these compact sets in simpler terms.
99 00:08:50,978 –> 00:08:53,518 This is called the Heine-Borel theorem.
100 00:08:53,986 –> 00:08:57,729 Therefore I hope I see you there and have a nice day. Bye!
-
Quiz Content
Q1: What is the $\varepsilon$-neighbourhood of $x$ for a point $x \in \mathbb{R}$?
A1: $[x, x+\varepsilon]$
A2: $(x, x+\varepsilon)$
A3: $(x-\varepsilon, x+\varepsilon)$
A4: $(x-\varepsilon, x+\varepsilon) \setminus { x }$
Q2: Which of the following sets is not a neighbourhood of the point $0 \in \mathbb{R}$.
A1: $\mathbb{R}$
A2: $(-2,2)$
A3: $[-2,11]$
A4: $[0,11]$
A5: $(-1,1) \cup [2,4]$
Q3: Which of the following sets in not open?
A1: $\mathbb{R}$
A2: $(-2,2)$
A3: $(0,\infty)$
A4: $(0,11)$
A5: ${0} \cup \left{ \frac{1}{n} \mid n \in \mathbb{R} \right}$.
Q4: Which of the following sets in not closed?
A1: $\mathbb{R}$
A2: $\emptyset$
A3: $[0,\infty)$
A4: $[0,11)$
A5: ${0} \cup \left{ \frac{1}{n} \mid n \in \mathbb{R} \right}$.
-
Last update: 2025-01
