general topology - muller
Contents
1. Se ts, functions and re lations 3
1.1. Sets 3
1.2. Functions 3
1.5. Relations 4
2. The integers and the re al numb ers 7
3. Pro ducts and c opro ducts 9
4. Finite and infinite s ets 10
5. Countable and uncountable s ets 12
6. We ll-ordered se ts 14
7. Partially ordered se ts, The Maximum Principle and Zorn’s le mma 16
8. Top ologic al space s 18
8.4. Subbasis and basis for a top ology 18
9. Order top ologies 20
10. The pro duc t top ology 20
10.4. Pro ducts of linearly ordered space s 21
10.6. T he copro duc t top ology 21
11. The s ubspace top ology 21
11.5. Subs pace s of linearly ordered s pace s 22
12. Close d s ets and limit p oints 24
12.3. C los ure and interior 24
12.10. Lim it p oints and is olate d p oints 25
12.14. C onve rge nc e, the Hausdorff prop erty, and the T1 -axiom 26
13. Continuous functions 27
13.6. Hom eomorphism s and emb e ddings 28
13.13. Maps into pro duc ts 29
13.17. Maps out of copro duc ts 30
14. The quotie nt top ology 31
14.1. Op en and c los ed maps 31
14.2. Quotie nt top ologie s and quotient m aps 31
15. Me tric top ologies 37
15.6. T he firs t countability axiom 37
15.13. T he uniform metric 39
16. Connec ted space s 40
17. Connec ted subsets of linearly ordere d s pac es 42
17.5. Path conne cte d s pac es 43
17.9. C omp onents and path c omp onents 43
17.12. Lo c ally c onnec te d and lo cally path conne cte d s pac es 44
18. Compact s pac es 47
19. Compact s ubspace s of line arly orde red space s 50
19.6. C ompactne ss in metric s pac es 51
20. Lim it p oint com pactnes s and se quential com pac tnes s 53
21. Lo c ally compact s pac es and the Ale xandroff compactification 53
22. Countability axioms
23. Se paration Axioms 58
24. Norm al space s 60
25. Se cond countable regular s pac es and the Urysohn me trization the ore m 62
25.1. An e mb edding the ore m 62
25.5. A universal s econd countable re gular space 63
26. Completely regular s pac es and the Stone– ˇCe ch c ompac tification 65
26.6. T he Stone– ˇCe ch c ons truction 65
27. Manifolds 68
28. Re lations b etween top ological spac es 69
Contents
1. Se ts, functions and re lations 3
1.1. Sets 3
1.2. Functions 3
1.5. Relations 4
2. The integers and the re al numb ers 7
3. Pro ducts and c opro ducts 9
4. Finite and infinite s ets 10
5. Countable and uncountable s ets 12
6. We ll-ordered se ts 14
7. Partially ordered se ts, The Maximum Principle and Zorn’s le mma 16
8. Top ologic al space s 18
8.4. Subbasis and basis for a top ology 18
9. Order top ologies 20
10. The pro duc t top ology 20
10.4. Pro ducts of linearly ordered space s 21
10.6. T he copro duc t top ology 21
11. The s ubspace top ology 21
11.5. Subs pace s of linearly ordered s pace s 22
12. Close d s ets and limit p oints 24
12.3. C los ure and interior 24
12.10. Lim it p oints and is olate d p oints 25
12.14. C onve rge nc e, the Hausdorff prop erty, and the T1 -axiom 26
13. Continuous functions 27
13.6. Hom eomorphism s and emb e ddings 28
13.13. Maps into pro duc ts 29
13.17. Maps out of copro duc ts 30
14. The quotie nt top ology 31
14.1. Op en and c los ed maps 31
14.2. Quotie nt top ologie s and quotient m aps 31
15. Me tric top ologies 37
15.6. T he firs t countability axiom 37
15.13. T he uniform metric 39
16. Connec ted space s 40
17. Connec ted subsets of linearly ordere d s pac es 42
17.5. Path conne cte d s pac es 43
17.9. C omp onents and path c omp onents 43
17.12. Lo c ally c onnec te d and lo cally path conne cte d s pac es 44
18. Compact s pac es 47
19. Compact s ubspace s of line arly orde red space s 50
19.6. C ompactne ss in metric s pac es 51
20. Lim it p oint com pactnes s and se quential com pac tnes s 53
21. Lo c ally compact s pac es and the Ale xandroff compactification 53
22. Countability axioms
23. Se paration Axioms 58
24. Norm al space s 60
25. Se cond countable regular s pac es and the Urysohn me trization the ore m 62
25.1. An e mb edding the ore m 62
25.5. A universal s econd countable re gular space 63
26. Completely regular s pac es and the Stone– ˇCe ch c ompac tification 65
26.6. T he Stone– ˇCe ch c ons truction 65
27. Manifolds 68
28. Re lations b etween top ological spac es 69
