# An introduction to Point-Set-Topology Part - II

- Institution: IIT Bombay
- Department: Mathematics
- Teacher: Prof. Anant R Shastri
- Level: Undergraduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111101160)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111101160)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

This course begins with the assumption that the audience has attended the part-I. We cover all important advanced topics such as various notions of compactness in metric spaces, Ascoli's theorem, paracompactness, compactifications, Urysohn's Metrization theorem, Stone-Weiertrass theorem, totally disconnectedness, etc. Checking whether any of these topological properties in product invariant occupies a considerable amount of our time. A brief introduction to Topological Dimension Theory, discussion of function-space topologies and the Ordinal topology as a rich source of counter examples are the added features. Finally the course ends with a brief introduction to topology of manifolds in which we discuss the 1-dimensional case completely and outline the 2-dimensional case. The content of this course will be useful for any body who wishes to study deeper aspects of analysis or topology or in applying topological tools anywhere else in Mathematics or Physics, etc. It is certainly useful in understanding the two courses that I have offered on Algebraic Topology on NPTEL portal.

Intended audience: Students from various streams which include some modern mathematics such as B. Sc., M. Sc., Students, Ph. D. , B. Tech. and M. Tech. who have seriously attended PST-I or have acquired the basic knowledge the content offers from elsewhere, will be able to benefit from this course.

Prerequisites: Students are supposed to have attended the part-I of this course from NPTEL or acquired the basic knowledge the content the part-I offers from elsewhere.INTENDED AUDIENCE: Students from various streams which include some modern mathematics such as B. Sc., M. Sc., Students, Ph. D. , B. Tech. and M. Tech. who have seriously attended PST-I or have acquired the basic knowledge the content offers from elsewhere, will be able to benefit from this course.

## Lectures

1. [Lecture 1 : Preliminaries from Banach spaces](https://invisiblenetwork.ai/learn/nptel/111101160/edadef85b902) (YouTube)
2. [Lecture 2 : Differentiation on Banach spaces](https://invisiblenetwork.ai/learn/nptel/111101160/1c620371bff1) (YouTube)
3. [Lecture 3 : Preliminaries from one-variable real analysis](https://invisiblenetwork.ai/learn/nptel/111101160/fc3697616665) (YouTube)
4. [Lecture 4 : Implicit and Inverse function theorems](https://invisiblenetwork.ai/learn/nptel/111101160/9b5271856d24) (YouTube)
5. [Lecture 6 : Compact Hausdorff spaces](https://invisiblenetwork.ai/learn/nptel/111101160/27587cd34a30) (YouTube)
6. [Lecture 7 : Local compactness](https://invisiblenetwork.ai/learn/nptel/111101160/df603657661a) (YouTube)
7. [Lecture 8 : Local Compactness-continued](https://invisiblenetwork.ai/learn/nptel/111101160/1cffd8ef3bb9) (YouTube)
8. [Lecture 9 : The retraction functor k(X)](https://invisiblenetwork.ai/learn/nptel/111101160/5a358b56b339) (YouTube)
9. [Lecture 10 : Compactly generated spaces](https://invisiblenetwork.ai/learn/nptel/111101160/38d083281f49) (YouTube)
10. [Lecture 11 : Paracompactness](https://invisiblenetwork.ai/learn/nptel/111101160/fd924c5279bb) (YouTube)
11. [Lecture 12 : Partition of Unity](https://invisiblenetwork.ai/learn/nptel/111101160/85f6e0fc2c57) (YouTube)
12. [Lecture 13 : Paracompactness continued](https://invisiblenetwork.ai/learn/nptel/111101160/4746d2ed62cf) (YouTube)
13. [Lecture 14 : Paracompactness continued.](https://invisiblenetwork.ai/learn/nptel/111101160/663252418f83) (YouTube)
14. [Lecture 15 : Various Notions of Compactness](https://invisiblenetwork.ai/learn/nptel/111101160/818996168368) (YouTube)
15. [Lecture 16 : Total Boundedness](https://invisiblenetwork.ai/learn/nptel/111101160/caceb420c316) (YouTube)
16. [Lecture 17 : Arzel`a- Ascoli Theorem](https://invisiblenetwork.ai/learn/nptel/111101160/859dc997c8d6) (YouTube)
17. [Lecture 18 : Generalities on Compactification](https://invisiblenetwork.ai/learn/nptel/111101160/e354dba5bd8c) (YouTube)
18. [Lecture 19 : Alexandroff’s compactifiction](https://invisiblenetwork.ai/learn/nptel/111101160/c488e221e4bd) (YouTube)
19. [Lecture 20 : Proper maps](https://invisiblenetwork.ai/learn/nptel/111101160/d2e47a7c0b67) (YouTube)
20. [Lecture 21 : Stone-Cech compactification](https://invisiblenetwork.ai/learn/nptel/111101160/5b755361224a) (YouTube)
21. [Lecture 22 : Stone-Weierstrass’ Theorems](https://invisiblenetwork.ai/learn/nptel/111101160/d65894a0b34c) (YouTube)
22. [Lecture 23 : Real Stone-Weierstrass Theorem](https://invisiblenetwork.ai/learn/nptel/111101160/67388d9c11c3) (YouTube)
23. [Lecture 24 : Complex and extended Stone-Weierstrass theorem](https://invisiblenetwork.ai/learn/nptel/111101160/53735344dadf) (YouTube)
24. [Lecture 26 : Urysohn’s Metrization theorem](https://invisiblenetwork.ai/learn/nptel/111101160/6cedd5c6448d) (YouTube)
25. [Lecture 27 : Nagata Smyrnov Metrization theorem](https://invisiblenetwork.ai/learn/nptel/111101160/96002dfb7c1a) (YouTube)
26. [Lecture 28 : Nets](https://invisiblenetwork.ai/learn/nptel/111101160/23e4fa5e6e37) (YouTube)
27. [Lecture 29 : Cofinal families subnets](https://invisiblenetwork.ai/learn/nptel/111101160/4f9d117dc4db) (YouTube)
28. [Lecture 30 : Basics of Filters](https://invisiblenetwork.ai/learn/nptel/111101160/457ce283f733) (YouTube)
29. [Lecture 31 : Convergence Properties of Filters](https://invisiblenetwork.ai/learn/nptel/111101160/8a0e8e28b1ad) (YouTube)
30. [Lecture 32 : Ultrafilters and Tychonoff’s theorem](https://invisiblenetwork.ai/learn/nptel/111101160/d63c773baea4) (YouTube)
31. [Lecture 34 : Wallman compactification](https://invisiblenetwork.ai/learn/nptel/111101160/3de610d14f42) (YouTube)
32. [Lecture 33 : Ultraclosed filters](https://invisiblenetwork.ai/learn/nptel/111101160/417ecaa465df) (YouTube)
33. [Lecture 35 : Wallman compactification-continued](https://invisiblenetwork.ai/learn/nptel/111101160/881bdf6307a6) (YouTube)
34. [Lecture 36 : Global Separation of Sets](https://invisiblenetwork.ai/learn/nptel/111101160/135742d15f9a) (YouTube)
35. [Lecture 37 : More examples](https://invisiblenetwork.ai/learn/nptel/111101160/848c0c758317) (YouTube)
36. [Lecture 38 : Knaster-Kuratowski Example](https://invisiblenetwork.ai/learn/nptel/111101160/b92866db3721) (YouTube)
37. [Lecture 39 : Separation of Sets-continued](https://invisiblenetwork.ai/learn/nptel/111101160/60a165035d82) (YouTube)
38. [Lecture 40 : Definition of dimension and exampels](https://invisiblenetwork.ai/learn/nptel/111101160/2a011694e8ea) (YouTube)
39. [Lecture 41 : Dimensions of subspaces and Unions](https://invisiblenetwork.ai/learn/nptel/111101160/12295e88042a) (YouTube)
40. [Lecture 42 : Sum theorem for higher dimensions](https://invisiblenetwork.ai/learn/nptel/111101160/6f8fd521765c) (YouTube)
41. [Lecture 43 : Analytic Proof of Brouwer’s Fixed Point Theorem](https://invisiblenetwork.ai/learn/nptel/111101160/a30a52c15ec5) (YouTube)
42. [Lecture 44 : Local Separation to Global Separation](https://invisiblenetwork.ai/learn/nptel/111101160/f2ab47e85622) (YouTube)
43. [Lecture 45 : Partially Ordered sets](https://invisiblenetwork.ai/learn/nptel/111101160/55a7e2fbd2e9) (YouTube)
44. [Lecture 46 : Principle of Transfinite Induction](https://invisiblenetwork.ai/learn/nptel/111101160/55856faebbc3) (YouTube)
45. [Lecture 47 : Order topology](https://invisiblenetwork.ai/learn/nptel/111101160/81b86c66e068) (YouTube)
46. [Lecture 48 : Ordinals](https://invisiblenetwork.ai/learn/nptel/111101160/fd3a29e1cf73) (YouTube)
47. [Lecture 49 : Ordinal Topology Continued](https://invisiblenetwork.ai/learn/nptel/111101160/2366482eb89c) (YouTube)
48. [Lecture 50 : The Long Line](https://invisiblenetwork.ai/learn/nptel/111101160/912a40272afe) (YouTube)
49. [Lecture 51 : Motivation and definition](https://invisiblenetwork.ai/learn/nptel/111101160/ba106f6e939e) (YouTube)
50. [Lecture 52 : The exponential Correspondence](https://invisiblenetwork.ai/learn/nptel/111101160/344220780bf6) (YouTube)
51. [Lecture 53 : An Application to Quotient Maps](https://invisiblenetwork.ai/learn/nptel/111101160/7cb03dfffe70) (YouTube)
52. [Lecture 54 : Groups of Homeomoprhisms](https://invisiblenetwork.ai/learn/nptel/111101160/2444867216e9) (YouTube)
53. [Lecture 55 : Definition and Exampels of Manifolds](https://invisiblenetwork.ai/learn/nptel/111101160/2a2b5ce30bae) (YouTube)
54. [Lecture 56 : Manifolds with Boundary](https://invisiblenetwork.ai/learn/nptel/111101160/fc52caf61aed) (YouTube)
55. [Lecture 57 : Homogeneity](https://invisiblenetwork.ai/learn/nptel/111101160/aac14574e562) (YouTube)
56. [Lecture 58 : Homogeneity continued](https://invisiblenetwork.ai/learn/nptel/111101160/73ed3a665f9a) (YouTube)
57. [Lecture 59 : Classification of 1-dim. manifolds](https://invisiblenetwork.ai/learn/nptel/111101160/ed3f8bdfdd26) (YouTube)
58. [Lecture 60 : Classification of 1-dim. Manifolds-cont.](https://invisiblenetwork.ai/learn/nptel/111101160/81fd49097a74) (YouTube)
59. [Lecture 61 : Surfaces](https://invisiblenetwork.ai/learn/nptel/111101160/266a25a6ced4) (YouTube)
60. [Lecture 62 : Connected sum](https://invisiblenetwork.ai/learn/nptel/111101160/6c810d04b357) (YouTube)