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Open & Closed Sets
Lesson 32 of 51 • 2 upvotes • 7:30mins
Asim Anand
Open and closed sets.
Mathematics Optional for UPSC - Real Analysis
51 lessons • 5h 27m
1
Overview
2:06mins
2
Natural Numbers, Algebraic Structure and Order Structure [Chapter 1]
8:16mins
3
Inverse Operations, Integers and Rational Numbers, their Algebraic and Order Structures
8:01mins
4
Fields of Numbers, Set of Real Numbers as a Complete Ordered Field
9:00mins
5
Intervals, Supremum and Infimum
8:27mins
6
Properties of Supremum & Infimum
5:42mins
7
Greatest (smallest) member of set bounded above (below)
5:02mins
8
Bounded & Unbounded Sets
3:48mins
9
Solved Examples
13:58mins
10
Order Completeness of set of Real Numbers - I
5:42mins
11
Order Completeness of set of Real Numbers - II
4:52mins
12
Equivalent descriptions of Order Completeness property, Dedekind Property
5:28mins
13
Equivalence of Dedekind Property & Order Completeness Property
5:41mins
14
Set of Real Numbers as a Complete Ordered Field - Summary, Archimedean Property
7:03mins
15
Archimedean Property (cont'd.)
5:56mins
16
Archimedean Ordered Field, Denseness Property of R
7:49mins
17
Denseness Property of R (cont'd)
3:58mins
18
Solved Examples
11:19mins
19
Modulus of a Real Number - I
6:49mins
20
Modulus of a Real Number - II
8:01mins
21
Principle of Mathematical Induction, Bernoulli's Inequality [Chapter 1 conclusion]
5:09mins
22
Neighbourhood & Deleted Neighbourhood of a Point, Examples [Chapter 2]
6:48mins
23
Examples cont'd.
6:38mins
24
Properties of Neighborhoods
5:14mins
25
Properties of Neighborhoods (cont'd), Limit Point of a Set
6:07mins
26
Limit Point of a Set (cont'd), Derived Set, Examples
5:57mins
27
Examples (cont'd)
5:02mins
28
Bolzano-Weierstrass Theorem
6:10mins
29
Bolzano-Weierstrass Theorem (cont'd), Theorem II
8:20mins
30
Theorem III, Theorem IV
5:27mins
31
Theorem V
4:03mins
32
Open & Closed Sets
7:30mins
33
Theorems concerning Families of Open & Closed Sets - I
4:51mins
34
Theorems concerning Families of Open & Closed Sets - II
5:01mins
35
Theorems concerning Families of Open & Closed Sets - III
5:08mins
36
Examples of Open & Closed Sets
5:39mins
37
Interior, Exterior & Boundary of a Set, Examples
6:47mins
38
Theorems on Interior of a Set - I
5:54mins
39
Theorems on Interior of a Set - II
6:04mins
40
Adherent point and Closure of a Set
5:12mins
41
Theorems on Closure of a Set - I
5:46mins
42
Theorems on Closure of a Set - II
5:04mins
43
Theorems on Closure of a Set - III
5:40mins
44
Isolated Points, Discrete Set, Dense Sets
6:59mins
45
Illustrations
5:34mins
46
Cantor Nested Interval Theorem
10:57mins
47
Cover of a Set
5:58mins
48
Compact Set
5:01mins
49
Properties of a Compact Set
7:56mins
50
Heine - Borel Theorem
7:28mins
51
Properties of a Compact Set (cont'd.) [Chapter 2 conclusion]
6:41mins
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