 Hello, welcome to NPTEL NOC, an introductory course on point sector power here, part 2. I am Anand Arshastri, retired emeritus fellow from IIT Bombay Department of Mathematics. I have a team of teaching assistants here, Professor Ankum Tikken Singh from Nehru in Shilong, Professor Arjala and Bhopang also from Nehru Shilong, Oankar Javedkar from IIT Bombay and Sushil Singla from Shivanadar University. So as name suggests, this is the part 2 of the point sector power here, point sector power here, part 1 was also a NPTEL NOC course which was introduced in the last semester. You can have a look at the content page here, bibliography and content page, chapter 1 on differential calculus on Banach spaces, then compactness and separation axioms, there are compactness and partition affinity, other notions of compactness, compactification, metrization, nets and filters and global separation axioms and then Hurwitz-Walman dimension theory and one more chapter named it is not shown here unfortunately. Very important chapter manifolds. So the point is that the second part naturally assumes that the participants of this course are fairly familiar with the content of part 1. We actually do not assume that you have attended part 1, you may have learnt that part on your own somewhere else from through books or some other course but if you have attended part 1 lecture and have understood a good percentage of that material then sailing with this part 2 will be very easy for you. So such people will benefit much better than people who have not attended part 1. The course is delivered in 60 modules of approximately 30 minutes age just like many other NPTEL courses. We expect that learners will be able to cope up with 5 such modules per week that is like a 2 and half hours contact hour. An important aspect of mathematical studies is to keep a continuity. If you do leave something out somewhere in between it is most likely that you will have difficulty in understanding from later chapters, later portions. It can happen immediately also or it may happen little later also. Quite often it happens immediately so that is what mathematics is. So the key here is keep the continuity, continuity of studies. In any case you have your YouTube, you are free to use them in your own free time so that is the whole advantage of having these recorded lectures. In order to have the students in this respect, in this respect means work in keeping touch with the whole thing. We have a discussion portal in which all participants can raise their queries. Suppose you have not understood something despite having tried. You can try to get these difficulties squared by the team of these TAs. So they are quite enthusiastic, enthusiastic about teaching. They will be constantly looking into these queries. The common portal is there for all of you to put your questions and then they will be responding to them also. The set of lecture notes for part 1 as well as for part 2 will be made available right in the beginning of the course. So that will also help you to understand the material. Along with this, there are also a number of pre-planned live interactive sessions directly with me. With more enthusiasm from the participants from your side, additional interactive sessions can be arranged that is courtesy NPTEL team of course. Let me give you a little elaborate description of the course details. We come to the old spirit namely part 1 whatever we started with, namely what to impact the basic topological ideas and training which will be very useful to people who may not want to study topology itself later but may want to do other subjects also like differential topology, differential geometry, algebraic topology. These are directly topological areas whereas lot of topology is used elsewhere like first of all in analysis for example. Nowadays topology is used by physicists also. A cursory glance at the content which I have shown earlier should confirm this claim. So the entire course along with part 1 and part 2 will not take you much deeper in any particular aspect but it will give you a wider and more fundamental you know training that is the whole idea. Apart from covering all standard topics in topology at this level I mean we have also covered some material which is not available in standard textbooks. The entire course is divided into 12 chapters varying length of course okay not exactly this much portion and so on. Depending on a certain thematic cohesiveness we have classified them chapterized that is all. In finally many chapters many ideas intermingle they are not completely independent after all the entire mathematics is cohesive work. In the first introductory chapter just like in part 1 we discussed something about Rn okay and this time little more namely Banach spaces. So we discussed differential calculus on Banach spaces. This is not a standard topic in topology nor it can be found in elementary differential calculus. Usually it is included or covered in some advanced analysis books. Since this result uses a little more topology which we have already developed in part 1 we feel that we have chosen the right place for it here. Chapter 2 is about compactness and separation axioms and heavily depends upon materials that we have covered in part 1 that is the whole idea of giving you at least the notes of part 1 also. We mix compactness and house darkness which form a central concept these two are central concepts in all topological discussions. Apart from that we have introduced some new concepts such as locally compactness as well as compactly generated spaces. Chapter 3 is devoted to study of veracompactness and partition affinity which is one of the very very powerful tool in topology. In chapter 4 we introduced various notions of compactness such as sequential compactness, countable compactness etc. and their interrelationships especially for metric spaces. We also discussed characterization of compact subsets of a complete metric space in terms of totally boundedness something like what we have done Hannibal theorem and so on. Finally we present the well-known Ascoli's theorem which deal with Bolzano-Oestras property in function spaces you know real valued or complex valued function spaces which are very very important in analysis particularly. In chapter 5 we discussed compactification in general and then two specific ones namely one is called Alexander's compactification or one-point compactification the other one is Stonetsch's compactification. Discussion on related concept to one-point compactification namely the concept of proper maps is included. We then move on to the full discussion of Ston-Weistras theorem. Ston is common here. Ston is a Ston-Weistras theorem they are different of course. The Ston-Weistras theorem you must be knowing that it is about approximating continuous functions by polynomials that is the standard Weistras theorem that will be widely generalized by Ston. In chapter 6 we present Eurydor's matrization theorem and Nagata-Smirnov matrization theorem two of the important ones there are many other versions which we cannot cover. So these things brings out the close relationship with metric spaces and para-compactness especially the Nagata-Smirnov. In chapter 7 we study the concept of nets and filters. Each of them is a vast generalization of the concept of sequences. These are devices invented to do the job of sequences in the absence of a metric. There are several interesting follow-ups of the study. Among them an easy proof of Tyknoff's theorem is work mentioning. Of course we will cover that. Also with the help of filters we could then discuss another important compactification. This is the third one namely due to Wallmann. This is called Wallmann compactification. The other ones were Alexandrov and Stoncheck. In chapter 8 we introduce another not so standard topic namely global separation properties as compared to the local separation you know horse dorseness, regularity, normality and so on. The three separation axioms horse dorseness, regularity, normality are essentially local in nature. So in this chapter we introduce certain global versions of them. They are very close to them but they are stronger than that stronger than local version which is just a step first step towards the general theme of globalization that has to come in the next chapter namely dimension theory. So this is like a precursor preparation for the next chapter dimension theory. Essentially we are giving a detailed account of the concept of zero dimensionality without introducing the term dimensionality you know that is hidden here but this chapter is as as I told you it is a precursor to the next chapter which is about the dimension theory. So I am closely following the book of this Khuravish and Wallmann for this chapter and the next one. In chapter 9 we introduce the study of topological dimension theory due to Khuravish and Wallmann. Our aim here is a modest one for instance we shall prove that the topological dimension of every non-empty open set in Rn is equal to n for deeper aspects of this theory as well as comparison with other dimension theories such as there are many other dimension theories such as Lebesgue covering dimension the reader may look into the book of Khuravish and Wallmann okay we are only covering a very little amount of the material covered in this book. Chapter 10 is devoted to the study of order topology and the space of ordinals with a view to introduce a number of counter examples in an economical way. On the way we discuss some basic set theory such as well ordering and the principle of trans-finite induction just as a matter of you know recalling for ready reference we are not going to teach you the elementary set logic okay logical aspect of the set theory that we are not going to okay so these things are useful everywhere in higher mathematics so we just recall them also here after that we go to order topology and that is our motivation is to introduce a lot of interesting counter examples. In chapter 11 we discuss the problem of putting topologies on function spaces and discuss the compact open topology in some detail. We then present two applications of this the first one is a result due to Michael on quotient spaces quotient maps in the second one we discuss the question whether the group of homeomorphisms of a space forms a topological group under the compact open topology or not so this part is based on a talk I gave on this topic in a meeting at ISI Kolkata in 2019 to celebrate 60th birthday of our close friend Parameshwaran Shankaran from IMSC. In the last chapter number 12 we come to the study of central objects in topology namely manifolds after laying down some foundational results on general topological aspects of manifolds we devote some time in complete classification of one-dimensional manifolds and the last part is a sketch of the classification of two-dimensional compact manifolds so especially this part though it comes at the end we would like to make it a joyful experience like playing with you know piece of paper and so on okay you might have heard about Mobius band and such an important aspect of the course is the set of time bound assignment released each week they are not only meant for 30 percent of the total marks you know 30 percent 70 percent meant for final exam 30 percent is meant for these assignments but these assignments they are essential part of the course imparting problem solving skill they range from quite easy to sufficiently hard of course they will depend on the portions covered in the course up to that point see they are weekly assignment so end of first week you will get some assignment based on the first week things or maybe something calling part one that that's the kind of arrangement is there okay learning the correct solutions through interaction even after submission will greatly help in solving subsequent assignment as well as in the final exam okay so take care of that just because you have submitted these maybe it's wrong maybe it's not wrong the learning process is not over it's only beginning okay here is a list of references some of them are listed because I have borrowed material from them some of them are there because as a student I got familiar with them and liked them and like them even today some of them are there because I would like you to look into them for your further study some of them I might not have studied myself for the course itself you need not look anywhere else I am going to give you full notes as I declared so here are the references Armstrong's basic topology is a very popular book Lagundu G is topology that's the universal book this is the book many of generation you know topology is from our generation learnt topology from this very respected book okay so Douglas Harris this is a paper Khorevij Malman two or three chapters fully I am referring to that book this is my ex-kali's very nicely written book like a this is like a guide to another important book Kelly Kelly's general topology from which I learnt my own topology as that so there is another paper here then Michael's paper Milner's paper and this is Booh emergence of this also another paper Booh Christ topology is also very popular this book I found it very nice foundations of topology by Vainay Patti so he for example covers the Banach space you know differentiability inverse function theorem Banach space is the only book I have seen then there are books of you know Rodin, Rogar's okay my own three books are there basic complex analysis elements of different topology basic algebraic topology there is a little Booh very elementary Booh and very readable Booh Satish Chiralee and Hari Krishna Vasudeva on metric spaces then there is a paper here and this book is for example it's like a bible for me this is also one of my favorite book in when I was in MSA then this teen and seabag it's like a this is like a reference you know it's like a dictionary of all count examples also this Willard's book similar to Dugunji's book quite respected book okay another important aspect of this entire course is that some of these symbols are frozen what does it mean that I am not going to use these symbols for anything else like if I say X is a topological space X could be a set X could be a subset and so on this is general notation right A, B or little X and so on but these symbols they are all in Euler font are space of real numbers C space of complex numbers Q space of rational number 33 ring of integer this n is set of natural numbers j power n minus 1 to plus 1 the interval taken n times the product minus 1 to plus 1 so similar just i is a closed interval 0 1 dn is closed unit disk in Rn okay for example if d equal to 1 then this will be a closed interval from minus 1 to plus 1 is an open interval here Sn is a unit sphere in Rn plus 1 Pn is n dimensional real projective space don't worry if you don't know these things but these things have been introduced in part 1 also C Pn is a complex projective space then I have this standard notation here because quite often I want to treat R the real numbers and complex numbers together so I have this standard notation k for both of them so whether it is real numbers or complex numbers depends upon the context if it is very specific then I will write R or C otherwise it is k it means that you can take either R or C okay so thank you for your attention and once again I welcome you to this course thank you very much.