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 ℝ^{n}

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 ℝ^{n}

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 ℝ^{n} are closed and bounded (Exercise 17)

The Heine–Borel Theorem: Closed and bounded subsets of ℝ^{n} are compact


2020.12.10:
html
; Exercises 18, 19
Continuous realvalued 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 twoelement 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 T_{2} property and the weaker T_{1} 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 𝔽_{2}^{ℕ}, the Zariski topology and Tychonoff topologies are the same.

The compactness theorem for propositional logic is the Tychonoff Theorem for 𝔽_{2}^{ℕ}

There is no compactness theorem for secondorder 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 