 Welcome to NPTEL NOC course and introductory course on point set topology part 1. So that should also tell you that there is a part 2 also planned and that is true. So I am Anand R. Shastri, retired Emirates fellow from Department of Mathematics at Bombay. I am well supported by my team members, Professor Ankom Tkeng Singh from Nehu, Shilong. Hello everyone, I am A. Tkeng Singh. I am from Nehu, Shilong. And Professor Ardelen Bhopang, again Nehu, Shilong. Hello everyone, I am Adeline Mary Bhopang. Omkar Javdekar from IIT Bombay. Hello everyone, I am Omkar Javdekar from IIT Bombay and I welcome you to this course. Sovik Wander from ISI Bangalore. Hello everyone, I am from ISI Bangalore. And Vinay Sipani, IIT Madras. Hello all, this is Vinay Sipani. Thank you all. I should first of all thank NPTEL portal as well as the staffs from Nehu. I have got this very nice opportunity to speak to you people. And also it was a very enjoyable experience to work with NPTEL staff here at IIT Bombay. So thank you all for that. And now let me tell you about this course in general and how to cope up with this course when you want to study this course. So this course is open to everybody. Everybody who wants to learn is welcome. It comes in two parts as I have told you. This part is part one that will be presented right now in the semester. The first part requires a little bit of mathematical maturity and a taste for abstract mathematics. Apart from some basic calculus, we assume that the participants are exposed to a certain amount of real analysis, some linear algebra and a little bit of group theory. If we need any special or specific results from these areas, we shall be very happy to recall them during the course. If we feel all students above the BSc or BTEC level should be able to successfully and beneficially do this part of the course. So of course they should be knowing their calculus and the basic things. That's all. So it goes without saying that we assume that the participants for the second part would have already attended this first part or would have acquired this much of knowledge from somewhere else. So part one is more or less compulsory from part two. We will not have any patience or anything to do with people who don't know the part one course. In the part two course. So that is what the general plan is. Now the course is divided into 60 modules of approximately 30 minutes each. We expect the learners will be able to cope up with 5 such modules every week. That is like two and half hours of lecture courses. An important aspect of all mathematical studies is to keep up the continuity. If you do leave something in between, it's most likely that you will have difficulty in understanding the later part of the course. In order to help the students with this aspect, we will have a discussion portal. This is all available for all NPTEL courses in which all participants can raise any kind of queries regarding the course and whatever help they want and so on. A team of TAs will be constantly looking into these queries and responding to them. Along with this, especially for this course, there is a possibility of number of pre-planned live interactions directly with me as well as some of the TAs. We will have one such meeting right in the second week of whenever the course begins. The subsequent meeting will depend upon the response we get in terms of attendance as well as how actively you participate in the interaction. So please make use of these facilities to the best of your abilities. So, we hope to make this course a joyful learning experience. I once again welcome all of you to this course. The contents of part one is divided roughly into five chapters. This is only for convenience that is nothing more than that. The first four weeks, I have just called this introduction. So what it is, I will tell you a little more. The second part, the two weeks, I have called it as new topological spaces out of the old. So this is a technique how to create new topological spaces. Smallness properties takes the third chapter for another three weeks. Then there are some so-called largeness properties that we will discuss for two more weeks. In the last week, we will give you some glimpse of topological groups and topological vector spaces. So this is roughly the division of this entire course. Out of this, what are the kind of things that you get? So I will give you a little more elaboration on this topic. One of the big area of mathematics where topology is frequently used is analysis. Also a lot of topological theory is based on what we see, what we experience in metric spaces. So the natural thing for us to start with is the norm linear spaces just like our prototype namely Euclidean space the so-called r r square r r power n and so on. And more generally, drop out the linear part and just go to the metric part. Those ways are called metric spaces and then and those are the ones which give you a large class of topological spaces. So the course begins in the first week with a brief introduction to norm linear spaces and metric spaces. Unlike many standard books and courses, unlike I am telling you, we immediately introduce the general concept of topological spaces right in the second week though most of the time we work with metric spaces themselves. We cover most of the important results on metric spaces such as cantor intersection theorem, Panark contraction mapping principle, Bayer's category theorem and Lebesgue's covering Lemma during the first four weeks. The fifth and sixth way we begin the earnest study of topological spaces. Now the metric spaces and norm linear spaces are at the background now creating new spaces out of the world. Concepts such as induced and co-induced apologies with emphasis on special cases such as subspaces, quotient spaces, product spaces. These are introduced here and studied and they will be again and again occurring within the course all over. In the third chapter that is in seventh and eighth and ninth week, we shall begin the study of topological properties. There are broadly classified into two classes, smallness property and largeness properties. According to the nature of the properties that I will not go into now detail. So that that is the portion for seventh, eighth and ninth week. We shall study what are called as path connected spaces, connected spaces, then compact spaces, Lindelof and first countability, second countability, separability. So then some interrelation between these concepts. Through a thorough discussion of the persistence of this property, what do you mean by persistence? Suppose you take a subspace, suppose you take a quotient space, will the same property will hold there also? Suppose you take a product of two spaces, if each of them has that property, will the product will have that? These are called persistence properties. This will be discussed very thoroughly in this slide. In the fourth chapter, the same kind of study will be done for what are called as largeness properties. In each some of these largeness properties are precisely fresh space, hostile space, regular space, normal space, etc. And then again interactions, interrelations between them. Here, two important results are Urizon's lemma, it is called Urizon's, it is actually characterization. Similarly, T's characterization of normal spaces. So the normal spaces are the sort of heroes here with the Urizon's and T's characterizations taking care of them, important results in this part. Finally, in the last week, we briefly introduced the concept of topological groups and topological vector spaces. The first topic lays foundation for studies of new groups. And the second one is a useful introduction to functional analysis, wherein a lot of topology is used anyway. Time bound assignments are not only meant for evaluation, they are part and parcel of the course, especially imparting problem solving skills. So we want to, we want all of you take seriously this part of the course. So this is part and parcel of the course. The level at which these things are there, it will be quite easy also, but a moderate level is also there. Some more difficult things are also there, especially as the core proceeds, you may find those things more and more difficult, especially if you have ignored the earlier thing. If you have learned earlier thing properly, so throughout you may find the whole thing easy also. So learning the correct solutions through interaction or interactive sessions, asking questions, getting the answers and so on, that will help you not only in solving the problems later on, the following assignment, they will be very, very useful at the final exam also. That is the only way to consolidate things that you have learned properly. Okay. Here is a list of references. Some of them are listed because I have borrowed materials from them. Some of them are there because as a student I got familiar with them and like them even today. Some of them are there because I would like you to look into them for your further study. Actually, I might not have studied them. Okay. Of course, you need not look anywhere just for this course itself. Okay. I am going to give you the full notes right from the first day itself. So that will be in PDF format, not in this Beamer file. Okay. So here is some of these things I will like you to go through this one. Anyway, this list will be also there for you later on. So Armstrong's basic topology. Dugundi is a classical one, topology, universal book style. Purvich and Walman, this is an ancient book, which is a very, very serious book on dimension theory. My own colleague, ex-colleague now, they are all retired. K. D. Yoshi, wonderful book, introduction to general topology. I own, you know, this Kelly's book is a hard book. This was the book, famous book when we were students and I am quite thorough with this book. Then this is some paper I have quoted because I have presented some part of this one. Then this E. Michael is a wonderful topologist, topologist. He has done a lot of research work. So I have referred to one of his theorems here. Then there are these books, George Moore's book. This is, this is not a book. This is an article which will give you some historical background for open sets and closed sets. Recently I have been familiar with this, Wainey-Patties Foundation of Topology. It's a nice book. I have not studied this, but I like this book. Then our function analysis by Rudin. My own book on complex analysis and elementary differences of topology and algebraic topology. Then there is this neat little book by Sirali and Vasudeva on metric spaces. So I have borrowed some material from here also. It is a wonderful book. Lastly, but not leastly, this George Simmons book is the book I studied for my MSc. And this is a very wonderful book. Finally, there is this book by Steen and Seeback, Counter Examples in Topology. This is like a dictionary of all counter examples. This is a wonderful book. Nowadays it is available online. And the present edition is much, much larger than the edition which I had seen as a student in 30 years back. And there is this Stephen Willard's General Topology, which is a standard book in many universities. But I have myself not studied this one. So that is roughly the thing. Then finally, this I will repeat again when the course starts again and again. So these are the some standard Euler forms I am using for nothing else during the course. So R, this Euler font R is always used for real number. C is space of complex number. Q is space of rational number. Z is ring of integer. N is the set of natural number. This j, this j is minus 1 to plus 1, the interval j power n is an Cartesian coordinate space. You know taken n times. And then the common notation K standing for either R or C. Quite often we will not work with real number or complex number. Both of them it will be true. So I will use the notation K. This I always introduce the closed interval 0, 1. Dn, the closed ring is Sn, the unit sphere in Rn plus 1. Pn and Cpn are real and complex projective spaces. This you may not know right now, but later on I will introduce them. So these are the standard notation I am going to use. So thank you for your attention. I hope you will all enjoy this course. Thank you.