1
Introduction
▶
ColDoc
[2G2]
Copyright
[009]
2
Notations
3
Fundamentals
▶
3.1
Logic
3.2
Set theory
3.3
Relations
3.4
Order relations
3.5
Functions
3.6
Elementary functions
3.7
Projecting to the quotient
3.8
Natural numbers in ZF
3.9
Well ordering
3.10
Cardinality
3.11
Operations on sets
3.12
Combinatorics
4
Natural numbers
▶
4.1
Induction
4.2
Recursive definitions
4.3
Arithmetic
4.4
Ordering
4.5
Z-F and Peano compatibility
4.6
Generalized induction, well ordering
4.7
Frequently, eventually
5
Groups, Rings, Fields
6
Real line
▶
6.1
Neighbourhoods
6.2
Frequently, eventually
6.3
Supremum and infimum
6.4
Limits
6.5
Upper and lower limits
6.6
Approximation of irrational numbers
6.7
Algebraic
7
Sequences and series
▶
7.1
Sequences
7.2
Recursive sequences
7.3
Series
7.4
Generalized sequences, or “nets’
7.5
Generalized series
8
Topology
▶
8.1
Neighbourhood; adherent, isolated and accumulation point
8.2
Examples
8.3
Generated topologies
8.4
Compactness
8.5
Connection
8.6
Nets
8.7
Continuity and limits
8.8
Bases
8.9
First- and second-countable spaces
8.10
Non-first-countable spaces
9
Miscellanea
▶
9.1
Polygons
9.2
Cantor set
10
Metric spaces
▶
10.1
Definitions
10.2
Topology in metric spaces
10.3
Quotients
10.4
Distance function
10.5
Connected set
10.6
Topology in the real line
10.7
Topology in Euclidean spaces
10.8
Fixed points
10.9
Isometries
10.10
Compactness
10.11
Baire’s Theorem and categories
10.12
Infinite product of metric spaces
10.13
Ultrametric
10.14
P-adic ultrametric
10.15
Circle
11
Dimension
12
Normed spaces
▶
12.1
Norms in Euclidean space
12.2
Isometries
12.3
Total convergence
12.4
Norms of Linear application
12.5
Norms of Matrixes
12.6
Minkowski sum
12.7
Mathematical morphology
13
Semicontinuity, right and left limits
▶
13.1
Semi continuity
13.2
Regulated functions
13.3
Sup transform
14
Continuity
▼
14.1
Continuous functions
14.2
Uniformly continuous functions
14.3
Lipschitz and Hölder functions
14.4
Discontinuous functions
15
Convex functions and sets
▶
15.1
Convex sets
15.2
Convex function
15.3
Real case
15.4
Additional properties and exercises
16
Riemann integral
17
Differentiable functions
▶
17.1
Higher derivatives
17.2
Taylor polynomial
17.3
Partial and total derivatives, differentials
17.4
Implicit function theorem
17.5
Constrained problems
18
Limits of functions
▶
18.1
On Ascoli–Arzelà’s Theorem
19
Power series
▶
19.1
Sum and product, composition and inverse
19.2
Exp,sin,cos
19.3
Matrix exponential
20
Analytic functions
21
Curve
▶
21.1
Closed curves
22
Surfaces
23
Ordinary Differential equations
▶
23.1
Autonomous problems
23.2
Resolution
23.3
Qualitative discussions
23.4
Envelope
23.5
Linear equations
23.6
Matrix equations
24
Written exams and pseudo–exams
▶
24.1
Functional equations
24.2
Vector Fields
A
Bibliography
14 Continuity
[14J]