Polytropy

David Pierce, 2020 güz dönemi

Notlar (çoğunlukla İngilizce ve pdf)

• 2020.10.22: 10 pages, size A5, portrait orientation. Exercises 1–6.

  • Continuity, neighborhoods, and open sets in ℝ

  • Definition of a topological space

  • Continuity, neighborhoods, and open sets in arbitrary topological spaces

  • Examples: discrete, trivial, and cofinite topologies

• 2020.11.01: Türkçe, 7 sayfa, A5 boyu, dikey. Ödevler hakkında.

• 2020.11.05: 14 pages, size A6, landscape orientation

  • Neighborhoods of ∞ and −∞

  • ℝ ∪ {∞, −∞} as the topological space ℝ^*. Its subspace ℕ ∪ {∞} [or ℕ^*]

  • Closed sets

  • In a topological space, the neighborhoods determine the open sets

  • The subspace topology

  • Homeomorphisms

• 2020.11.12: 10 pages, size A6, landscape orientation; Exercises 7–11

  • In ℝ :

    • closedness and boundedness

    • Extreme Value and Intermediate Value theorems as entailing that the continuous image of a closed and bounded interval is a closed and bounded interval

    • Exercise 9: the continuous image of an arbitrary closed and bounded set is closed and bounded

  • Limit points of subsets of arbitrary topological spaces

  • The order topology on a linearly ordered set

  • The order and subspace topologies on a subset of a linearly ordered set

• 2020.11.19: 8 pages, A6, landscape; Exercises 12, 13

  • Metrics and examples on ℝⁿ

  • Topologies defined by metrics

  • Bases of a topology

  • The product of two topological spaces

• 2020.11.26: 9 pages, A6, landscape; Exercises 14, 15

  • More on Exercise 9

  • Cantor Intersection Theorem on ℝⁿ

  • Countably infinite and uncountable sets

  • Cantor’s theorem that always A ≺ ℘(A)

  • The Cantor set

• 2020.12.03: 8 pages, A6, landscape; Exercises 16, 17

  • More on Exercise 9

  • Open coverings of topological spaces

  • Compact topological spaces

  • The continuous image of a compact set is compact (Exercise 16)

  • Compact subsets of ℝⁿ are closed and bounded (Exercise 17)

  • The Heine–Borel Theorem: Closed and bounded subsets of ℝⁿ are compact

• 2020.12.10: html; Exercises 18, 19

  • Continuous real-valued functions on compact metric spaces are uniformly continuous

  • Connected topological spaces

  • Connected components (correction to Exercise 19: two elements of a topological space will be in the same connected component if some connected subset of the space contains them both)

• 2020.12.17: 11 pages, A6, landscape

  • sufficient condition for being a basis of a topology

  • Zariski topology

  • two equivalent metrics on the Cantor set

• 2020.12.24: 9 pages, A6, landscape; Exercises 20, 21

  • Cantor set and Zariski topology for the countably infinite power of the two-element field

• 2020.12.31: 10 pages, A6, landscape; Exercises 22, 23

  • More on the topology of the power set of the natural numbers

  • The Tychonoff topology

• 2021.01.07: 8 pages, A6, landscape; Exercises 24, 25

  • More on the Zariski topology and compactness

  • The Hausdorff or T₂ property and the weaker T₁ property

• 2021.01.14: 8 pages, A6, landscape

  • The compactness of the spectrum of a commutative ring follows from the Prime Ideal Theorem

  • That theorem follows from Zorn’s Lemma

  • The Tychonoff Theorem follows similarly

• 2021.01.21: 8 pages, A6, landscape; exercise 26

  • On 𝔽₂^ℕ, the Zariski topology and Tychonoff topologies are the same.

  • The compactness theorem for propositional logic is the Tychonoff Theorem for 𝔽₂^ℕ

  • There is no compactness theorem for second-order logic

Ödevler

1 (29 Ekim)

Exercises 1–4

2 (5 Kasım)

Düzeltmeler

3 (12 Kasım)

Exercises 5, 6

4 (19 Kasım)

Exercises 7–9

5 (26 Kasım)

Exercises 10, 11. In Exercise 10, let Ω be ℝ; otherwise for (a) and (b) there are counterexamples! Eke bakın

6 (3 Aralık)

Exercises 12, 13

7 (10 Aralık)

Exercises 14, 15

8 (17 Aralık)

Exercises 16, 17

9 (24 Aralık)

Exercises 18, 19 (yukarıdaki düzeltme vardır)

10 (31 Aralık)

Exercises 20, 21

11 (7 Ocak)

Exercises 22, 23

12 (14 Ocak)

Exercises 24, 25

13 (4 Şubat)

Exercise 26

Exercise 1 & 2 çözümleri

Eğer bana gönderdiğiniz ödevleri görmediğimi düşünüyorsanız, bana söyleyin. Aşağıdaki tabloda:

• 0 veya boş = verilmemiş

• 1 = verilmiş ama yanlış veya eksik

• 2 = tam ve doğru