ZFC Axioms
The axioms of set theory
Categories
Math / Set Theory
Ordered Pairs
Subset relations of sets and the definition of ordered pairs
Binary Relations
Definition of Binary Relation
Operations on Binary Relations
Inverse and Composition of Binary Relations
Functions
Basic Definition of Functions
Operations on Functions
Inverse and composition of functions, surjective and injective functions
Retraction and Section
Properties of Surjective and Injective Functions
Unions and Intersections
Unions and intersections of sets
Sum of Sets
Sum of Sets (Disjoint Union)
Product of Sets
Product of Sets
Properties of Products
Partial products, associativity and distributivity
Equivalence Relations
Definition and properties of equivalence relations
Examples of Equivalence Relations
Examples of equivalence relations, saturation of equivalence relations, isomorphism theorems
Order Relations
Definition and properties of order relations
Monotone Functions
Operations on ordered sets and monotone functions
Elements in Ordered Sets
Maximum, minimum, maximal, and minimal elements of ordered sets
Directed Sets
Directed sets and lattices
Filters, Ideals, and Galois Connections
Filter and ideal
Ordinals and Well-Ordered Sets
Definition of well-ordered sets, motivation for ordinals
Properties of Well-Ordered Sets
Rigorous definition of ordinals and properties of well-ordered sets
Axiom of Choice
The axiom of choice and its equivalents
Order Relations Between Ordinals
Order relations between ordinals and the rigorous definition of cardinals
Cardinals
Definition of Cardinal number
Operations on Cardinals
Operations on cardinal numbers
Natural Numbers and Infinite Sets
Definition of natural numbers and properties of infinite sets
Limits
Inverse limit and direct limit
Math / Algebraic Structures
Algebraic Structures
Binary operations defined on sets
Semigroups, Monoids, Groups
Definitions of semigroups, monoids, and groups
Grothendieck Groups
The Grothendieck group and the definition of integers
Group Homomorphisms
Definition and properties of group homomorphisms, kernels and images
Quotient Groups
Normal subgroups and quotient groups
Isomorphism Theorems
The isomorphism theorems
Direct Products of Groups
Direct product of groups
Restricted Sums
Restricted sum of groups
Free Products
Free product and universal property
Abelian Groups
Free abelian group, tensor product
Group Actions
Group action
Definition of Rings
Definition and basic properties of rings
Quotient Rings and Ring Homomorphisms
Quotient ring and ring isomorphism theorems
Field of Fractions
Localization, ring of fraction, prime ideal
Modules
Definition of module
Math / Algebraic Topology
Homology
Definition and Properties of Simplices