Introduction
[100]
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ColDoc
[2G2]
Copyright
[009]
1
Notations
2
Fundamentals
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2.1
Logic
2.2
Set theory
2.3
Relations
2.4
Order relations
2.5
Functions
2.6
Elementary functions
2.7
Projecting to the quotient
2.8
Natural numbers in ZF
2.9
Well ordering
2.10
Cardinality
2.11
Operations on sets
2.12
Combinatorics
3
Natural numbers
▶
3.1
Induction
3.2
Recursive definitions
3.3
Arithmetic
3.4
Ordering
3.5
Z-F and Peano compatibility
3.6
Generalized induction, well ordering
3.7
Frequently, eventually
4
Groups, Rings, Fields
5
Real line
▶
5.1
Neighbourhoods
5.2
Frequently, eventually
5.3
Supremum and infimum
5.4
Limits
5.5
Upper and lower limits
5.6
Approximation of irrational numbers
5.7
Algebraic
6
Sequences and series
▶
6.1
Sequences
6.2
Recursive sequences
6.3
Series
6.4
Generalized sequences, or “nets’
6.5
Generalized series
7
Topology
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7.1
Neighbourhood; adherent, isolated and accumulation point
7.2
Examples
7.3
Generated topologies
7.4
Compactness
7.5
Connection
7.6
Nets
7.7
Continuity and limits
7.8
Bases
7.9
First- and second-countable spaces
7.10
Non-first-countable spaces
8
Miscellanea
▶
8.1
Polygons
8.2
Cantor set
9
Metric spaces
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9.1
Definitions
9.2
Topology in metric spaces
9.3
Quotients
9.4
Distance function
9.5
Connected set
9.6
Topology in the real line
9.7
Topology in Euclidean spaces
9.8
Fixed points
9.9
Isometries
9.10
Compactness
9.11
Baire’s Theorem and categories
9.12
Infinite product of metric spaces
9.13
Ultrametric
9.14
P-adic ultrametric
9.15
Circle
10
Dimension
11
Normed spaces
▶
11.1
Norms in Euclidean space
11.2
Isometries
11.3
Total convergence
11.4
Norms of Linear application
11.5
Norms of Matrixes
11.6
Minkowski sum
11.7
Mathematical morphology
12
Semicontinuity, right and left limits
▶
12.1
Semi continuity
12.2
Regulated functions
12.3
Sup transform
13
Continuity
▶
13.1
Continuous functions
13.2
Uniformly continuous functions
13.3
Lipschitz and Hölder functions
13.4
Discontinuous functions
14
Convex functions and sets
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14.1
Convex sets
14.2
Convex function
14.3
Real case
14.4
Additional properties and exercises
15
Riemann integral
16
Differentiable functions
▶
16.1
Higher derivatives
16.2
Taylor polynomial
16.3
Partial and total derivatives, differentials
16.4
Implicit function theorem
16.5
Constrained problems
17
Limits of functions
▶
17.1
On Ascoli–Arzelà’s Theorem
18
Power series
▶
18.1
Sum and product, composition and inverse
18.2
Exp,sin,cos
18.3
Matrix exponential
19
Analytic functions
20
Curve
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20.1
Closed curves
21
Surfaces
22
Ordinary Differential equations
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22.1
Autonomous problems
22.2
Resolution
22.3
Qualitative discussions
22.4
Envelope
22.5
Linear equations
22.6
Matrix equations
23
Written exams and pseudo–exams
▶
23.1
Functional equations
23.2
Vector Fields
A
Bibliography
9 Metric spaces
[0MR]