1
ZFC Axioms
The axioms of set theory
2
Ordered Pairs
Subset relations and the definition of ordered pairs
3
Binary Relations
The definition of binary relations
4
Operations on Binary Relations
Inverse and Composition of Binary Relations
5
Functions
Basic Definition of Functions
6
Operations on Functions
Inverse and composition of functions, surjective and injective functions
7
Retraction and Section
Properties of Surjective and Injective Functions
8
Union and Intersection
Union and intersection of sets
9
Sum of Sets
Sum (disjoint union) of a family of sets
10
Product of Sets
Product of Sets
11
Properties of Products
Partial products, associativity and distributivity
12
Equivalence Relations
The definition and properties of equivalence relations
13
Examples of Equivalence Relations
Examples of equivalence relations, saturation, and isomorphism theorems
14
Definition of Order Relations
Definitions and properties of order relations
15
Monotone Functions
Operations on ordered sets and monotone functions
16
Elements of Ordered Sets
Greatest, least, maximal, and minimal elements of ordered sets
17
Directed Sets
Directed sets and lattices
18
Filters, Ideals, and Galois Connections
Filter and ideal
19
Ordinals and Well-Ordered Sets
Definition of well-ordered sets, motivation for ordinals
20
Properties of Well-Ordered Sets
Definition of ordinals and properties of well-ordered sets
21
Axiom of Choice
The Axiom of Choice and its equivalents
22
Order Relations Between Ordinals
Order relations between ordinals and the rigorous definition of cardinals
23
Cardinals
Definition of Cardinal number
24
Operations on Cardinals
Operations on cardinal numbers
25
Natural Numbers and Infinite Sets
Definition of natural numbers and properties of infinite sets
26
Limits
Inverse limit and direct limit