## Topoloji

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, the Zariski topology and Tychonoff topologies are the same.

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

• 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

Alıştırma AK AP BK CT IRB KD ME SeB SiB SKa SKo
1 1 1 1 2 1 1 1 0 1
2a 1 1 1 2 1 1 1 1 1
2b 2 1 1 2 1 1 1 1 1
2c 1 1 1 1 1 1 1 1 1
3 & 5 1 1 2 1 2 1 1 2
4 & 6 1 1 2 1 1 1 1
7(a) 2 2 1 2 2 2 1 2 2 2
7(b) 2 1 2 2 2 2 1 0 2 2
8(a) 2 2 2 2 2 2 2 1 2 2
8(b) 2 2 2 2 2 2 1 1 2 2
9 1 1 1 1 1 1 1
10(a) 1 1 1 1 1
10(b) 1 1 1 1 1 1 1
10(c) 0 1 1 1 1
11 2 2 2 2 2 2 1 2 2
12 1 1 2 2 2 1 1 2
13(a) 1 1 2 2 2 1 2
13(b) 1 1 1 1 1 1
14 1 2 1 1 1 1 1
15(a) 1 1 1 0 1 1
15(b) 2 2 2 2 2 2 2 2
15(c) 2 1 2 1 1 1 1 2
16 2 2 2 2 2 2 2 2 2
17(a) 2 1 2 2 2 2 1 2 2
17(b) 2 1 2 2 2 2 2 2 2
18 2 2 1 2 2 2 1 1 2
19 2 2 1 1 1 1 1 2 1
20(a) 2 2 2 2 2 2 2 2
20(b) 2 1 2 2 1 2 2 2
20(c) 1 0 0
21(a) 2
21(b) 2 1 1
21(c) 1 1
22 1 1 1 1 1
23(a) 2 2 2 2 2 2 1 2 2
23(b) 2 2 2 2 2 2
24(a) 2 2 1 2 1 1 1 2
24(b) 1 1 1 2 1 1 1 2
25(a) 2 2 1 1 1
25(b) 1 0 1 1 1
26 1 1 2
AK AP BK CT IRB KD ME SeB SiB SKa SKo