 Welcome to NPTEL NOC, an introductory course on point set topology part 2. Today we shall start a new topic topology of manifolds. Manifolds are central objects in the study of topology. Though the idea of a manifold can be traced back to Riemann in his work on the so called Riemann surfaces, of course Riemann did not call them Riemann surfaces obviously, a formal definition of this manifold may be attributed to Herman Weis. Its study is a must in any kind of higher mathematics and things that use higher mathematics like theoretical physics and so on. Our aim here is quite modest dealing with only few, very few salient topological aspects of topological manifolds as compared to deeper studies such as you know additional structures with PL manifolds, smooth manifolds, league groups and so on so forth, complex manifolds and so on. So, we are just taking topological manifolds which in some sense encompass all such special things ok. So, it is so general therefore, it is going to be somewhat weak statements are somewhat weak here ok, but they will be available for all those studies. So, both ways they are good in that way alright. First we shall study some topological aspects of manifolds in general. We shall then take up classification of one-dimensional manifolds. Next we shall merely outline the classification of compact two-dimensional manifolds. This is aimed at motivating you people to study other topological areas like algebraic topology and so on ok, especially the NPTEL course on algebraic topology there are two courses. So, this will be a motivation as well as this entire course will be a very good preparation for those courses finally ok. So, today module 50, 55 definitions and examples of manifolds. Fix an integer n, take a non-empty topological space by an n-dimensional chart for x, we mean a pair u, psi consisting of an open neighborhood u of some point x inside x and homeomorphism psi from u to Rn, u is an open subset of x, u to Rn on to an open subset of Rn. Often when I say homeomorphism it is not necessarily on to here, but on to some open subset you take that subspace topology there it is homeomorphism ok. Here I specifically mentioned that the image is open inside Rn that is very important here, u is open inside inside x does not mean that psi u is open inside Rn so that has to be hypothesis here ok that is the condition here. Now once you have defined a chart I am going to define an atlas ok these are borrowed from geography I think all these things go to Gauss ok. So by an atlas we mean a collection of u j psi j of chart ok, chart collection of chart what is the condition the union of all these domains of the chart must be the whole of x. If there is an atlas for x then we say x is locally Euclidean. In other words the chart gives you local Euclideanness of capital X at the point x if it happens at all the point then it is called locally Euclidean x is the local Euclidean. A chart u comma psi is called a chart at x naught if you can choose this psi to be such that psi of x naught is 0 this is a very special special definition I have cooked up so that I do not have to keep on saying that psi of x naught is 0 at x naught means the function is such that x naught is chosen as the origin going to the origin here. So also we keep writing psi as psi 1 psi 2 psi n these are n coordinate functions because psi is taking values inside r x. So these n component functions coordinate components of psi these are called local coordinate functions for x at the point x. The local coordinates remember not defined on the whole of x they are defined in a neighborhood of this little x or x naught. For n greater or equal to 1 I think n capital n is the same thing as n greater or equal to 1 here in t here x be a topological space we say x is topological manifold of dimension n if it is locally Euclidean in the above sense that is a class consisting of n dimensional charts now this n is occurring there all same thing same n. The second and third condition is just you know ordinary topological condition is host or space and second countable ok. So these two extra conditions report because we do not want to deal with non-host or spaces we do not want to deal with second non-second countable spaces because then we cannot do any you know analysis. Any countable discrete space is a zero dimensional space zero dimensional manifold do not confuse this one with the definition of zero dimensionality for a topological space which is just separable metric space ok. So this is a manifold definition we have got it it is a subset of that ok but we insist that it is countable discrete space here we do not want to take all arbitrary zero dimensional spaces second countable separable metric spaces which you have defined as zero dimension namely satisfying S2 ok. So that is the difference here it will satisfy S2 because it is discrete space but converse may not be true that is why we have to be careful here. So this will cover now n equal to zero case also ok and n equal to minus one is covered if you want namely when x is empty set that is allowed. We would like to include the empty space also as a topological manifold however there is no good way of assigning a dimension to it some authors prefer it to be dimension minus one just like in our dimension theory and some others put it as minus infinity. Indeed there are theories you know in the topology of manifold itself where it is better to treat the empty set as having any dimension whatsoever you want it is like zero polynomial ok zero polynomial usually people do not define the degree ok but it is better to keep zero polynomial having any degree it wants depending upon the context. Observe that once a chart view psi is you know exist at a point x not belonging to x means what now psi of x not is zero then we can choose a another chart v comma phi at x not such that this phi of v is the whole of r n. In fact this is some point going to some some open subset of r n may not be the whole of r n you can take a smaller set like this and you can make you have to change psi you have to change phi of v is the whole of r n. Therefore you could have assumed right in the beginning that all the charts have this property but that is not always convenient ok it is better to have as many of them the moment you fix up things there may be less chance that is all ok by composing with a translation we can assume first of all psi of x not is zero and then we can choose r positive such that an open ball of radius r is contained inside this open subset around zero right and then take v to be psi inverse of br0 then what happens psi itself is a homeomorphism from v to br0 but now br0 the whole of r n you have lots of homeomorphism you compose with them for example you can take x equal to x by r square minus x square this is a homeomorphism very nice homeomorphism from br0 ok ball of radius r open ball of radius r centered at 0 goes to the whole of r n for a nut last it is necessary to assume that the integer n is the same for all the charts ok do not change the n there of course if x is connected you are free to change also but you have to use a certain theorem which you have seen namely braver invariance of domain ok namely r n is not homeomorphism r n so you will not have much chance there because once different n's are used the n that occurs will be a locally constant function on x but x is connected therefore a locally constant function has to be constant so this n and m I cannot occur only one of them will occur ok so that is a minor point it is better to assume that you do not have like a line and disjoint union with a point such things we do not call it as a manifold that is all ok because line can be parameterized by open subsets inside r right itself is a single parameterization single point you have to take it as a discrete space ok single discrete space it is a countable discrete space therefore it is 0 dimension so do not mix up such a different dimension that is all it says alright there is the dimension of the chart at a point becomes locally constant and a locally constant function and a connected space is a constant function ok for a topological space that is locally Euclidean second countable and off-drawal ok condition the second countability why we choose that one is equivalent to many others what is the meaning of the under locally Euclidean say locally Euclidean implies remember it is what it is locally compact it is locally convex and so on right so lot of things in case it will be locally Euclidean means lot of things second countability condition is equivalent to many others such as metrizability para compactness plus connectedness ok all these things are equivalent so you find second countability is the most suitable for our purpose ok easy to understand the Eurydense metrization theorem that we have done tells you that every second countable t3 space is metrizable locally compact off-drawal space is rt3 space therefore they are metrizable so I can I could have put metrizability and then I can conclude that it is second countable ok since manifolds are locally Euclidean they are locally compact and hence regular ok metrizability is not enough separability you have to put ok therefore we obtain every manifold is metrizable in particular every manifold is para compact because we have proved metrizable space is para compact ok so to sum up started with you locally Euclidean plus saucedor then you can have just put that it is metrizable and separable then it is second countable or you can just say para compactness and connectedness that is also good enough so all these things are you know going one into the other in particular given any open cover U of a manifold X using local compactness we can first take a refinement V of U see I am now talking about this para compactness hence you have para compact you have partition affinity also I want to see something more here in the case of manifolds so what is that start with a manifold and an open cover take a refinement V which consists of open sets of closures are compact so here I am using local compactness such sets are called relatively compact see we have never used it say so many times we have used U such that U bar is compact right there is terminologies for relatively compact means that you could have used that terminology then we can get a partition affinity theta I subordinate to this V subordinate means what support of theta I closure support of theta itself is actually closure of all the point where in theta is not 0 so that support of theta is a closed set already that should be contained inside members of one of the members of V but members of V are such that their closures are compact therefore subordinate this theta I have compact support so that is what automatically this will imply that we will be subordinate to U also and the extra property support of theta is are all compact so this is not true in general for para-compact spaces okay we have a partition affinity which is subordinate to U that is all but I have extra hypothesis here namely support of theta is are compact all this will be quite useful now I am going to use this one so let us see some examples now if you have studied some differentiable manifolds inside Rn like a circle sphere parabolas ellipses ellipsoid and so on they are all manifolds okay take some time to see that right but I am not going to do that I am going to do only simpler examples okay so that is your calculus course full of manifolds there the boundary of the unit square as a subspace of R2 okay is a topological one fanifold even simpler if you take a triangle any triangle okay the boundary of that triangle consisting of three sides that will be a topological one manifold you may have learnt that this is not a smooth manifold inside R2 it is not a smooth manifold because you know triangles have corners the squares have corners and so on because of the corner point cost problems right if you do not know that do not worry they are all topological manifolds why because suppose you have corner like this okay what you can do you can this this line segment union this line segment is homeomorphic to this line segment right all that you have to do is turn this one this vertical thing make it horizontal like this okay you can write down your own you know use cost data find data formula and write down the homeomorphism from this union in fact you can try to homeomorphism from this first quadrant to the entire thing also okay so this is a nice illustration this picture tells you how to handle corner see here point here point here locally they are they are lines open sub open line segment they are no no problem but here what is it you have just straighten it out to get a homeomorphism into the interval that's all all right now let X be the union of two axis in R2 X axis or Y axis X axis and Y axis you can draw it anywhere you do not have to take the axis itself if you use any connected neighborhood of 00 how does it look like if you throw a 0 from you it will have four connected components okay therefore you can see that such a open neighborhood of you namely neighborhood U of this point 00 inside X axis the union of X is right cannot be homeomorphic to an open interval okay cannot have any chart covering 00 and hence fail to be a topological manifold all right so if at all you should have a homeomorphism to R because everywhere else except this point 00 the intersection point everywhere else you have neighborhood which are homeomorphic to open interval but at that point you don't have now I will give you some very specific example here which violates the house darkness we have put house darkness forcibly suppose house darkness is a consequence and you don't have to put that one now no it is not a consequence that means what I am going to construct a locally Euclidean locally one dimensional Euclidean space which is actually can take countable so it can second countable also or you can cut it off and make it compact also no problem but it will not be housed off okay let us see how let X be the set of all real numbers okay that is the real line together with one extra point that we shall denote by 0 quiddle we shall make X into a topological space as follows top your collection of all subsets A of X of the form A is B union C where B is allowed to be a empty or an open subset of R in the usual topology for empty set is also open subset of R in here but I have I want to emphasize this fact that it could be just empty why you are writing a B union C okay that is for first part what about C and C is either empty C could be also empty or C as the property the C is a subset both B and C are subsets of X right the first part set it is a subset of R itself that means 0 twiddle is not there now second part also says it is empty or 0 may be there you see 0 twiddle may be there so you intersect it with R that means what you are throwing away 0 twiddle but now you take union with 0 throw away 0 twiddle and put back this 0 that is now a neighborhood of 0 inside R is that clear what is the relation between 0 and 0 twiddle if you just ignore the other one then the rest of them looks like a real line if you ignore one of them both of them are there and they are so close to each other you cannot separate them the moment you take a neighborhood that neighborhood will intersect see this neighborhood is some open subset you throw away that point and put back this neighbor which is the other way around put back this point 0 twiddle that will become a neighborhood of 0 twiddle and vice versa okay therefore 0 and 0 twiddle cannot be separated by open set but they are different point so this is why it is called line with a double operation you can throw away many points like this at least you know several discrete set of points for each integer you can double them that will be a funny space but it will be locally Euclidean the same way you can not only one point you can put any finitely many points also like this the same property it is 0 twiddle 0 twiddle 0 prime and so on you can index them by another second so this is just idea but you can expand it to very large very complicated space also without the without the hypers house darkness you will get strange spaces okay which we do not want to deal with they do not occur in naturally it is only a consequence of our definition you know the deficiency of our definition therefore we have to put house darkness okay so I again describe this notice that the above rule describes all neighborhoods of 0 twiddle you may say first part describes all neighborhoods of 0 but this one describes all the neighborhood of 0 twiddle also namely take any neighborhood U of 0 okay throw away 0 and add the singleton add the 0 twiddle singleton so that is your C here if you take the intersection 0 twiddle goes away and 0 comes back here so that is a property alright so to verify that this is a topology all that you have to do is intersection finite intersection counter that we will leave to you okay let us go ahead now so this is one which we have studied a couple of days back the long line using our well-ordered in order topology and so on okay we know that long line is locally Euclidean okay it is housed off also what is it not it is not second countable it is not even first countable if you allow the omega but omega is not allowed inside the long line so it is first countable but it is not second countable so therefore long lines are not treated as many votes okay so that is an example so we should have you know sanctity for why these conditions are there okay we do not want to discuss long line on which you cannot do much analysis another type of non-example is obtained by taking disjoint union of manifors of different dimensions this I already dealt with like taking an open R and a singleton point outside or a line and a disjoint plane that things are not allowed we do not want to so for example such spaces are there if you go to algebraic geometry okay where in you know even larger class of objects are studied let us consider some examples of manifold that do not occur naturally as subspaces of rn so far our examples were all subspaces rn and counter examples were not subspaces of rn right now let us see something which does not occur naturally as subspaces but they are manifold so this is the one very very important example project space the foremost one in the n-dimensional real project space which we have introduced in part one but let me recall only a few aspects of this I hope you know if you don't know you can read it from there and so on so some few of them which perhaps are not there now it is there it is here now so first of all as the pn is at what is the definition of pn pn is the quotient of the unit sphere Sn inside rn plus 1 by the antipodal action what does that mean x and minus x are identified antipodal action means x goes to minus so look at the equivalence classes I can denote them by bracket x and take the quotient map q from Sn to Pn to be qx equal to this bracket x so put the quotient of all that is the way so space is defined over automatically this is a subjective of subjective map point it is a quotient map so it is continuous if Sn is compact therefore Pn will be compact okay so we have to see why it is locally Euclidean and Hausdorff after that we are happy because it is compact Hausdorff space already so it will be second countable also alright since Sn is compact Pn is compact now given any x belonging to S consider V to be the set of all points in Sn such that the distance of that points are less than square root of 2 from x so take the root 2 bar inside Sn plus inside rn plus 1 intersect with Sn that will be an open subset of Sn right now if you take x going to minus x y going to minus y okay under that this open subset will go to a disjoint open subset right therefore if you restrict the map q to V that will be a 1-1 map right so this q is actually homeomorphic to this V is actually homeomorphic to q under inside that okay q V will be homeomorphic because this is a neighborhood which is you can see that it is homeomorphic to an open ball inside inside rn I have taken open ball inside inside rn plus 1 intersecting with Sn so that is homeomorphic so that is a nice neighborhood so this will give you local description that that every point inside Pn has a neighborhood which is homeomorphic to an open subset of rn okay indeed this also tells you that the space is house door now because what you can do is let me see I will tell you something more take x and y disjoint a distinct point what does it mean plus minus x is not equal to plus minus y if x is not equal to y this bracket that classes are different means plus minus x is not so you have four different points now you take arbitrary four different points are right you take the least distance between all of them call that epsilon take epsilon balls around plus minus x epsilon or plus minus y all these four balls will be disjoint okay first of all this epsilon small minus of this plus minus that those will be disjoint from why are they are different because that is the way I have chosen this epsilon alright so that will show that when you take their images this b epsilon image q of that and q of this they will be neighborhoods of square x I mean this bracket x and bracket y so they will be disjoint so I included some exercise here show that a manifold is connected if and only it is path connected so these exercise very easy exercise but there is another exercise here this is not very central even if you do not understand this it is okay for some time okay so why this actually obsession with Hausdorff's and local Euclideanness okay so if you look at co-finite topology on R that is not a Hausdorff space okay but is it locally Euclidean you can ask alright so there are we have given a space like locally Euclidean it is not Hausdorff suppose Hausdorffness does it imply local Euclidean there are many such cases so like that you can go on asking questions so that is why I am discussing this one you can take a look at it and if you have difficulties you can come back to us okay so today let us stop here next time we shall introduce a larger class of manifold you are called manifold with boundary thank you