 This is an introductory talk on the NPTEL web course on topology here and on topology under this topic we have included five chapters we aim to give that is plan to give two or three lectures for each chapter. Let us see the motivation to start with what we mean by topology then we will give basic concept in topology and then we will introduce other notions we are all familiar with what we mean by real line and the Euclidean plane. And also we know if we take a subset say A a non-empty subset of R then we know that what do we mean by the set A is an open set that is recall A is said to be A is said to be on open set in R open set if for each x in A if for each element x here it is each x in A. So there exist a positive real number say we say there exist R greater than 0 such that the open interval x minus R x plus R is completely contained in A that is this open interval is a subset of A. So we have x is in the set A then we should able to find at least one R here that R is the radius this is the open interval we say that the center is x and radius R and this open interval is contained in A and if this happens for each x though we do not write R depends on x. So for each x the R may vary then such a case we say that A is on open subset of R for example if we take the open interval any open interval say A, B they are real numbers we will have A is strictly less than B now take any x in this is our set A here we take A is the open interval open interval A, B and we have taken take x in x belongs to the set A then sum x see the length of the interval open A to x and open x to B then find the minimum suppose this length is that is x minus A say R1 and B minus x say R2 both are greater than 0 because x is not equal to A x not equal to B. So this is greater than 0 now take R half of minimum of R1, R2 so if we take the half of this center is x radius is anything less than R. So then the open interval center at x radius R x minus R this is x plus R this open interval is completely contained in the set A then open interval x minus R x plus R is completely contained in A this is true for every x in the set A in our case our set A is open interval A, B suppose if x is even closer to B still we have this is our x still x B minus x is positive then half of the distance if this is our radius then this open interval will be so contained in open A, B. So I mean what we want is this R depends upon x so hence by our definitions the open interval A, B is an open set. So then it is easy to see that see for example open interval 1 comma A2 then say open interval 3 comma 4 then you take any x in either now take A equal to open interval 1 comma 2 union open interval 3 comma 4. If we take an x in A either x is in the in x is in open interval 1 comma 2 in this case they are disjoint so R x is in open interval 3 comma 4 so in any case we can find an R so is that this is our distance between x and this point is our R then the open interval x minus R this is x plus R is contained in open interval 1 comma 2 which is subset of A similarly if we take any x here we can find a suitable R sender at x and this interval will be completely contained in open interval 3 comma 4 which in terms subset of A. So this in set A is an open set so it is easy to see we will just give the statement so if we have a collection let us say capital A suffix alpha j is we say it is an index set meaning is alpha belongs to j, j is just any set so to start with we will assume that j is a non-empty set that means what for each alpha belongs to j we associate a set we have for each alpha we have a subset A suffix alpha of capital R further suppose each such alpha is an open set further suppose suppose each A alpha is open in R then the union then union of this set that is then union A suffix alpha belongs to j is also an open set in R just right the open in R again take x belongs to union A alpha alpha belongs to j then x will be in at least one A alpha that A alpha is open hence there exist an R greater than 0 such that the open interval x minus R x minus R x plus R will be a subset of that particular A alpha say that particular alpha say alpha naught then because A alpha naught is open then this is a subset of the union A alpha belongs to j so hence for each x in union that means x will be in at least one A alpha naught that A alpha naught is an open set that implies there exist on R greater than 0 such that open interval x minus R x plus R is contained in the union A alpha belongs to j hence this union is an open set. So we say that in such case A alpha is a collection of open sets then the arbitrary union because why we say arbitrary union mean A alpha is any arbitrary collection of open sets then the union is also open set this is true even when j is empty set when j is empty the union A alpha belongs to empty is the empty set and will see that empty set is also a open set under our definition what we mean by a set is open mean in R if it is a conditional statement A is open mean that is what if you assume that A non-empty then x belongs to A implies there exist R such that open interval x minus R comma x plus R is contained in A. So this statement is conditional so this when there is x the first statement is not true x is here first statement is x is in A that implies there exist some R such that x minus R x plus R is contained in A here when this statement is not true that is what imply P implies Q when the first statement is false whatever may be this second statement whether it is true or false this I mean P implies Q is true mean later we will see how this empty set so or we say that empty set sometime I say that take it as a we want that empty set should be open so that is same thing as telling that arbitrary union of open set is open including when the index set is empty similarly we can see that the intersection A alpha belongs to J when J is empty this will be the whole space. So the motivation is for example if you have A1, A2, A3 say in this case here countable collection countable means for each natural number we have a set A n then we take A1 first case union A2 A1 union A2 union A3 and so on here A1 the index set here it is 1 singled and 1 here index set is 1, 2 here index set is 1, 2, 3 and so on. So when the index set is smaller then the union here say call it J1, J2, J3 then J1 subset of J2 will imply A1 here subset of A1 union A2 this what we write union of A alpha alpha belongs to J2. So when the index it is smaller that is what empty set is subset of every index set so I mean that is the motivation to have the union is union A alpha alpha belongs to empty set is empty and when the intersection is the reverse A1 if we take the intersection A1, A1 intersection A2, A1 intersection A2 intersection A3 now the index set smaller here J1 then this is larger so similarly this when the hence when the index set is empty say it is like we have a collection of sets if you bring all together you will get a larger set if you do not bring together you would not get anything empty or we have the whole thing if you do not make it into pieces the whole space will remind you are not cutting into pieces. So that is the motivation to have this arbitrary union is empty when the index set is empty and the intersection A alpha alpha is empty equal to the whole space this idea is very important when we now define the topology. So in R we have seen that including empty set it is easy to see that the whole R is open reveal arbitrary union of open set is open also it is easy to see that A1, A2 open in R implies A1 intersection A2 is also open in R. So what we have is empty set is open in R the whole R is open in R arbitrary, arbitrary union of open sets in R is also open in R and the third is see here when we say arbitrary including the I mean index set empty but that we do not bother because of empty set in R we have already included and second is intersection finite intersection of open set is open for that enough to include A1, A2 or open sets in R implies A1 intersection A2 is also open in R this gives that J finite A alpha open for each alpha belongs to J that will imply intersection A alpha, alpha belongs to J is also open when alpha is empty sorry alpha belongs to J, J the index set is empty then the intersection will be the whole R which is already open so only we have to see when J is finite and non-empty so then use index N so if it is any finite set mean it is of that type A1, A2, An. So now A1 open assume that the result is true for N-1 so then we have A1 intersection A2 intersection An-1 is open An is open then we have proved that A1 intersection A2 is open this is exercise we can easily prove that A1, A2 are open in R implies A1 intersection A2 is open hence using that we can prove the intersection A1 intersection A2 up to intersection An is open that is now we call that as a collection tau the collection of all subsets of all subsets of R collection of all open subsets of R if we call that tau the collection of all open subsets in R then that collection is closed under arbitrary union and finite intersection this is the basic properties of R this can be I mean it is well known that if we introduce a notion called metric then the same concept we can extend to a metric space that will discuss when we introduce the other concept namely what is a basis so now let us introduce what is known as a topology so what is the idea in introducing the topology is first to start with R is study the concept of say open set then using open set we have the convergence what we call compact every if you can cover a set by a collection of open set from that we can extract finitely many members that will be sufficient the smaller collection will also cover that set A then that is called compact similarly we introduce what is called not connected mean there is a subset A such that A is non-empty not equal to the whole space so that A and A complement both are open such concept in a general metric space we call it connected all this notions depends only on open sets so instead of having a particular type of space like r power n later we will see even function spaces like c of 0 1 here we have a distance function notion see all this thing will come as a particular case of the space which we are going to define known as topology so what we mean by a topology is we will have this open set if A is a non-empty open set if we take x in A then we say that this A is a neighbor root of the point x so essentially what we have is we have a collection of neighbor roots that neighbor root has some these properties namely closed under arbitrary union and finite intersection see there are excellent books are available for this for topology so that is what this course is not a replacement of that available textbook that is not our aim under this NPTEL web course we have included notes meant for a one semester course say either for 40 hours or maximum 50 hours so this notes will be sufficient for one semester course then once you understand the basic concept that is why for each chapter first if a student wants to learn the subject with understanding the basic concept first listen for it to be for reading each chapter just listen this course one this whatever lectures given for that particular chapter then if you start if that student starts reading then you will have a maturity to understand the concept that is the aim of giving these lectures to three lectures for each chapter so the reference books are topology by mongers general topology by Kelly this is an excellent mongers is a students friendly I mean it is concepts are explained for in a students friendly way easy to understand whereas general topology by Kelly is a classical book it require maturity to understand other book are all equally they are all very good introduction to topology by K. D. Josie general topology by Willard topology by Duhunji so there are many other excellent books are available at least you have mentioned here some of the books now we will see what is the concept called topology we are going to replace the set are by on arbitrary set in fact x is any set in fact we do not even assume that it is non empty so x is any set any set so here then we know what is called power set of x the collection the collection of the collection of all subsets subsets of capital X all subsets that mean including empty set so now then from this collection we will have a sub collection that sub collection will have the properties namely that sub collection is closed under arbitrary union and finite intersection then that sub collection is called a topology on x so we have the power set p of x a sub collection tau so that notation is we read as tau take tau a sub collection I mean it is just a set it tau a subset of power set of x because here each element in tau mean it belongs to power set of x which is a subset of x hence we call tau a collection of subsets of x then this collection this collection tau this collection tau is said to be a topology on capital X if it satisfies if this collection satisfy if it satisfies the following we will just write it if it satisfies the following properties namely first that collection should contain the null set or we say empty set and the whole set x that mean this should belongs to tau this is the minimum requirement any tau empty set and the whole space x belongs to tau second it is closed under arbitrary union meaning is you take take j non-empty index set I mean meaning is j is just any set which is non-empty for each for each alpha belongs to j a alpha belongs to tau it is given that for each alpha belongs to j a alpha is a subset of x and that subset belongs to tau if it is so then this would imply closed under union union a alpha should also again this will be a subset this again belongs to tau and third if a1 this is symbol third a1 a2 belongs to tau implies a1 intersection a2 also belongs to tau if our tau a subset of power set of x has these three properties namely empty set x belongs to tau a alpha belongs to tau for each alpha implies union is in tau and a1 a2 is in tau implies a1 intersection a2 is in tau then this collection tau is called a topology on x so on why we call it a topology mean on the same x we can define more than one topology so now we will see some symbol example unlike in other metric spaces or other vector space and other notions here it is just we are starting with any set so it is easy to give construct example see to start with x is take on any set we can always construct x is any set okay just for we will avoid reveal tau contain only empty set so we avoid that we will assume that x is non-empty then now we example first call it say tau1 so we call it tau1 or say tau suffix t for reveal okay just we will write it here tau1 what is our minimum condition is empty set and x should be there so we cannot avoid that so empty set and x can we stop only this a smaller collection now whether as first condition is satisfied second says if a alpha belongs to tau where alpha is some a collection otherwise what this says that we have a sub collection of tau so but sub collection of here the tau is tau1 this has only two sets then sub collection will contain either empty set or singleton x or empty set and x whatever may be the union will be either empty or x it is to be it is close under arbitrary union again intersection same see what are the possible sets that is what empty set and x sing one set otherwise empty set and x so whatever may be the this only one set intersection this here intersection x with x empty set and empty set intersection x so this is reveal this is a topology this I will write in short topology on x the other extreme collect a this tau so either we call it tau suffix d for discrete or we say in this case just here tau suffix to the whole the other extreme which we call that is power set of x it contains the collection of all subsets then every empty set see what is empty set is also a subset empty set belongs to a subset it is in power set of x that is our tau2 and x is in subset of x it is in tau2 and if you take the union a alpha where each alpha belongs to tau2 mean it is a subset then union is also a subset so that will also belongs to tau2 similarly a1 is in tau2 a2 is in tau2 imply a1 intersection a2 is in tau2 seconds this is also a topology this is called known as discrete topology discrete topology we will call each a if it is belongs to tau tau is our topology then that a is called every member of tau is called open set that is our definition of open set so here every subset is an open set whereas here the only even x may be the set of real numbers but that case here empty set and x are the only open set here r is every set is a open set now we will see this is trivial example again take see if you want you can give any symbol example you can start with any set x say some finite set 1 2 3 take say call it tau3 we need empty set we need capital X say singleton 1 see that itself we see this set tau3 contain empty set x and singleton 1 it is closed under arbitrary union if you take any union either capital X or singleton 1 so both and intersection either it is empty or x or this is also a mean topology on x so it we can give any number of example say if we suppose empty set and the x 1 say 1, 2 now suppose we stop this whether whether is tau4 a topology on x so the obviously it is not closed under arbitrary union because if we take a1 singleton 1 belongs to our collection tau4 a2 singleton 2 belongs to tau4 but the union is not in tau4 so that mean it is not closed under arbitrary union for here what is our index set is j equal to 1, 2 then that is not closed mean it is not a topology topology on x so now again we will go back to any arbitrary x in this case we will assume that our x is a infinite set we will see why x any infinite set infinite that mean not a finite set infinite set and here we call it tau f f4 I mean finite this we see that this is called known as co finite topology go mean compliment should be finite here when you say that a subset a subset of x that mean a is in tau when we say is if and only if this I am writing for compliment of a is x difference a this is finite so note that already x is an infinite set so we want that first empty set belongs to here so when empty set suppose empty set if we take a equal to empty when suppose if we stop only here then compliment of empty set is the whole space which is infinite then empty set will not be here so it will not be a topology so the so but we need so to take care of that if a compliment is finite or the compliment is the whole space not finite mean a compliment should be the whole space then empty set now will come inside now we can see that this is here then this tau suffix here is a topology on x known as co finite known as co finite or finite compliment or finite compliment topology on x this will be we will come across many time for giving nice examples so now we will not see all the things see suppose I will give the quickly the how to prove that empty send x is empty set x belongs to tau f second suppose suppose a alpha belongs to some index at j we can assume that non-empty is a collection of a collection of members of tau f then what we have to prove is to prove or climb union a alpha belongs to j belongs to tau f but what is when some a will be here mean take the compliment two cases compliment either enough to prove it is finite if not finite it should be the whole space so take the our a here is take a equal to union of a alpha belongs to j then use the de Morgan law a compliment equal to compliment of this then by de Morgan law it is intersection a alpha compliment alpha belongs to j so now what is the if what can happen is we know already each a alpha compliment is either finite or the whole space suppose the worst thing can happen is a alpha compliment the whole space x for each alpha belongs to j then reveal each a alpha compliment is x then what will happen to a compliment the whole space so we are through otherwise a compliment for at least otherwise for at least one alpha not belongs to j a alpha not compliment is not equal to x that mean it should be a finite set this is finite then in this case then intersection will be a subset of that particular a alpha not compliment this is finite subset of a finite set is finite hence a alpha compliment is finite that implies our union is also in tau and intersection is almost reveal a one compliment finite e two compliment finite or the whole space then intersection of finite set is finite both are the whole space means the intersection is finite so similarly a one a two belongs to tau f imply a one intersection a two belongs to tau f hence tau f is a topology on x this topology is called co-finite topology in the next class we will see how to from your smaller collection which we call it basis from that how to construct a topology and other related concept thank you.