 Hello, welcome to module 50, NPTEL NOC course and introductory course on point set topology part 2. So continuing with the study of ordinals, we will now construct an example which is called long line. This long line is an example in usually in the study of manifolds that we are going to do in the last chapter here in this course. A line word is taken from some topological space which is homomorphic to let us say the whole of real line or an open interval. The long line is going to be something which is locally homomorphic to open interval that means every neighborhood has a every point has a neighborhood which is homomorphic to an open interval but the entire thing is not homomorphic to open interval first of all ok. That is not the whole point we will see that it is non-compact it is not even second countable that is the whole idea ok. It is a house door space. So, countability axioms are violated here. So, let us see why this is important I have told you let us see now the detail. Once again we have to deal with both the real line as well as the ordinal the 0 will become a conflicting point notation. So, restricting you know reserving the 0 for the ordinary 0 of the real numbers the 0 of the ordinal will be denoted by 0 quiddle ok. So, that is the first remark I have to make here. Also again we have ordered pairs of elements just like in the previous example of Ticross Planck. So, we will continue to use this x cross y for the ordered pair x y not to confuse it with interval x y ok. So, this notation x cross y will continue in addition we will have this 0 twiddle is the point the least element in the ordinal ok that is the difference ok. So, I start with again like just like in the Ticross Planck example you start with 0 twiddle omega open cross 0 omega open. So, both of them are open part here ok that is the difference, but do not confuse with there we have taken both of them as ordinals here. Now, this is the usual real number ok the open interval 0 to 1 both of them are ordered whenever you have a product of ordered a kind of posets product posets there is a lexicographic ordering namely an element of this one I will denote x cross t and another one y cross s where x is an element of the ordinals here and t is an element of the real numbers ok. So, y is first one is ordinal second one is a real number. So, you say this one precedes y cross s if it is known if x is less than y in the ordinals or if x is equal to y then t the second coordinate must be less than ok that is a lexicographic ordering depends upon look at the first letter if that is it earlier than the other word the words are arranged in this way in the dictionary that is why it is called lexicographic ordering or dictionary order. So, clearly if you define like this this brick is a total order on one l because if you have two elements here first you can compare x and y ok if x is less than y is going to you arrange them first of all then you go for if they are equal then you go for the second one that is all. So, it is always total order no problem because both of them are total order take the corresponding order topology as we have defined earlier by taking left trace and right trace right then this l comma tau this tau with less than or equal to is called long line long ray because there is a closer part here. So, it is a long ray then you remove 0 twiddle cross 0 the initial point here throw away that point that is called a long line it is just like closed interval 0 1 is a ray and then open interval 0 1 is a line ok that is all that is a reference between long ray and long line this is just a terminology has nothing nothing you know big deal here you could have interchanged the definition that is not good because this is a ray you can think of this also as a ray of course, but this is called long line ok now I will define one more thing here without much effort what I do I take two disjoint copies of l let us call them l 1 and l 2 and take the quotient space this you know font l you know Euler font l obtained by identifying the 0 1 with 0 of the second coordinate ok 0 1 cross 0 you can say and 0 2 cross 0 ok those two points are there right 0 because I have taken two I have taken two copies the copies I am saying this one ok you identify these two points where 0 twiddle I equal to 1 and 2 are copies of 0 ok we shall call l the longest line it is just if you have this longest long ray take two copies of that one copy reflected and the initial points are identified disjoint them that is the meaning of this identification that will be called long longest line ok I will justify this name later on when you justify this line long line this could be you know why this long line long longer line ok you will see that this longest there is no other bigger than line which you may call it as line that is the whole idea we shall implicitly identify the original space 0 comma 0 twiddle comma omega with the this coordinate space namely the second coordinate being 0 this is a real number ok so I will not have cumbersome notation every time cross 0 I do not have to write so that is my l not ok omega 2 cross omega cross the singleton 0 that whole thing is in the long line l ok so I am identifying this 0 twiddle comma omega with this subspace under the identification x goes to x cross 0 just to save ourselves a little bit of time and cumbersome notation that is all all intervals initial segment etc will be with respect to this new order on the product space inside l and inside the larger space for example now LP bar will denote all the elements inside q the q itself is you know denotes an ordered pair right which I have to write it is x cross t or something but all those q such that q is less than equal to p this will be the closed left ray right similarly closed right ray all those things with respect to this order you have to take not with respect to 0 omega or in the 0 0 infinity so they are themselves the sub sub partial order because the extended thing is a lexicographic order so here is a caution though we have used the Cartesian product notation for the underlying set the topology is not the product topology the topology is defined by using the lexicographic order right that is my tau brick it is tempting to describe the long line as obtained by taking disjoint union of closed ray intervals index over 0, omega that is the ordinal and identifying the end point x cross 1 with x plus 1 cross 0 starting point because see in the total order what is the next element to x cross 1 okay x cross 1 itself is not an element first of all we do not have we have to put suppose you put x cross 1 also that means in here in the product definition I suppose I take this closed interval here then what will be the next element to x cross 1 it will be x plus 1 in the notation of ordinal cross 0 right so you identify them then duplication will not be there so you can think of as if it is this point 0 is not here okay so that is a another way of defining many people when they give popular lecture they do that one that is what I want to tell is that is strictly speaking a wrong explanation so it is not quite good to explain it that way because that picture is good only in the initial stage namely 0, 0 plus 1, 0 plus 1 plus 1 and so on as soon as you limit ordinal let omega and so on the picture will be complete failure okay so there you have to strictly follow this rule okay better define it this way take the lexicographic ordering take the order of the college okay so note that for any x belonging to L0 the closed interval x, x plus 1 inside L is order preserving the homeomorphic to the closed interval 0, 1 if I take x cross 0 to x cross 1 open that is a nothing but x cross 0 comma 1 open the next point will be precisely x plus 1 cross 0 so you put that so this interval okay interval as an element inside capital L here curly L is order preserving homeomorphic to the closed interval 0, 1 okay if you throw away this point this point it is open interval 0, 1 by the very definition so this is the one which used in the in the in this wrong explanation of course okay but this much is correct but by using that globally you will get a wrong picture so that picture will be good only in parts not full picture. Second thing is the fundamental property of L that we are interested in from which many other properties follow okay is this property A capital A for each P inside L0 that means what is some x cross 0 okay I am just having a simpler notation P belonging to L0 not equal to 0 to L okay something beyond that we have look at the left ray LP bar closed left ray this is order preserving homeomorphic to the interval 0, 1 okay so let us prove this one for instance we can then immediately conclude that I am telling that this is the fundamental property once you have this one you can have this A prime sometimes I will use this A prime so this can be immediate derived from A let us see how we can conclude immediately that for each P prime which is now P cross T the T is now not 0 okay this LP prime bar is homeomorphic to again closed interval 0, 1 by the way homeomorphic you know I say all that you have to see is that it is homeomorphic some interval closed interval all closed intervals are homeomorphic to each other this we have seen several times okay for how do you do that this LP prime bar is nothing but LP bar union this vertical line segment right P cross 0, T up to T we have to go up to T right so this is already homeomorphic to closed interval 0, 1 okay that P cross P P cross 0 is a common point and this is homeomorphic 0, T the usual interval in the real line we can put them together you will get a homeomorphism 0 to T plus 1 instead of 0 to 1 okay and then you can rescale it okay so to get a homeomorphism from LP prime to the entire of 0 to 1 again so this you can just divide by 1 by 1 plus 2 alright so as a consequence it follows that this L is path connected by the way why this is homeomorphic to this one this is again the same thing so you see you have to just put them together one line another line another line another line and so on for each P belonging to L0 it is a countable union alright so countable union of closed intervals can be again made into a closed interval okay so how that is what that is what you have to show fine right so anyway so here as a consequence it follows that L is path connected because first of all you can join every point to 0 twiddle can be joined to every point inside L you see remember our order this 0 omega has lots of discrete points now those things have disappeared because on each point you have a line vertical line and the end points are identified end point of the top point in the bottom point next top on point they are identified they are they are near they are not identified I have deleted the top point in my definition so they are close to each other so this is what is happening here so so what happens is L is path connected because everything can be joined to some x cross 0 from x cross 0 you can go vertically to join it to any x cross t so as a consequence of a it follows that as a as well as a prime it follows that L is path connected path connected is connected okay not only that it is locally homeomorphic to an open interval in R right except the starting point 0 twiddle cross 0 at that point we have neighborhood which are homeomorphic to half closed intervals okay once something is homeomorphic to a closed interval a to b everything in the middle in the between strictly inside there will have neighborhood homeomorphic to open interval only the end points have a problem right here the other end is not a problem you can keep going further and further but the starting point will have a problem that is half that is why in the definition of long line we are deleting this point that is all okay so let us prove property a we have not yet proved that one I indicated already suppose the property a is not true for some x belong there okay what does that mean so if you take the set of all points such that this a property is not true that is a non-empty set so non-empty set will have least element so let y be the least element of all such elements if y equal to some z plus 1 that means y is a successor for some z no z inside L0 then we will Lz bar is homeomorphic to 0 1 that we have seen because up to Ly bar is homeomorphic okay because this is a least element for which this is not true see what it means just see you have got a this is not true and you have then you have take the least element okay so if y itself is z plus 1 z will be smaller than that that means it is true for that that means we have a homeomorphism from Lz bar to 0 1 okay then we can rescale it and as we have seen earlier okay and go all the way up to y also that will be contradiction okay so if it is not y not that plus 1 means y is a limit or in all right so that is the harder case what happens suppose y is a limit or in L what does it mean we have seen that there exist a strictly monotonically increasing sequence x n in L0 which converges to y okay so now it is a question how to see for all these x n the hypothesis will be true right a will be true only thing is for the limit what happens that is what question right so we may assume that let us say we have we have can always add some more points x0 is 0 greater from a we have for each this capital a we have for each xn okay n greater than 1 there is an order pizzerian homeomorphism hn from Lxn bar to 0 1 okay so all these x n's are smaller than s that is why this is possible now you take h of xn plus 1 h of h of n plus 1 of operating upon xn equal to a n okay for each n greater than equal to 1 so I am define this a n okay choose an order preserving homeomorphism alpha 1 from 0 1 to 0 half and for n greater than equal to 2 alpha n from having defined up to this one you take a n minus 1 a n minus a n is defined like this right a n minus 1 to 1 to 2 power n minus 1 minus 1 divided by 2 power n to 2 power n minus 1 minus divided by 2 power n okay so this is of length 1 by 2 power n define h you can always rescale right any two closed intervals are homeomorphic so take you choose this homeomorphism order preserving homeomorphism define h from now Ly to 0 1 by patching up all these alpha n composite hn's namely I will define h of p to be alpha n composition of p for p between xn minus 1 xn x x0 I am starting 0 0 okay 0 then x1 x2 and so on between xn minus 1 xn okay if p is there you define it by this method okay on the end points these functions coincide okay so this we can easily see now that up to s you have no problem okay sorry up to y how what what I should take h of y h of y to it as 1 remember if this is between xn it will never be this function will never be equal to 1 because it is always something 2 power n minus 1 by 2 power n so 1 is 1 is not yet covered you take h y equal to 1 it is easily verified that this is an order preserving homeomorphism the whole point is that all these 2 power n by 2 power n as n tends to infinity will convert to 0 convert to 1 okay so that is why if you define h y equal to 1 continuity comes bijection if you take open part okay here throw away y it is already a bijection and so you have come you have just you know concatenated one after another homeomorphism so it will be homeomorphism when you take one more point so both these elements convert to t converges to 1 then this y this in h of that will convert to 1 so you will get a homeomorphism so t sorry when p converges to y okay h of p as p converges to y will convert to 1 that is why this is continuous once you have a continuous bijection okay then you can show that it is a homeomorphism there is no problem therefore a is true so I recall I I took two different cases namely this y is a limit ordinal not a limit ordinal why is a limit ordinal okay so there are two cases we have proved that part a so from this one the rest of the topological aspects will be followed notice that argument we have used above is nothing but the principle of transfinite induction though we have put it in the language of proof by contradiction we could have easily put it as follows property a is true for 0 twiddle plus 1 that is easy to verify okay suppose it is true for all x less than y then the construction of h as above okay shows that it is true for y that step is of course we have to do okay assuming is less than is true for less x less than y okay then you prove it for y so that then it will follow that is true for all okay so here we have used a different language so we have what a suppose is not true and then take a least upper one in other words what I have used we have used the well ordering principle there you see right so we do not have to go to principle of transfinite induction that is all so this L is clearly Hausdorff okay there is nothing to prove here because every order topology is Hausdorff L is not second countable this is the point we have come now because it has uncountably many mutually disjoint open sets singleton x cross 0 1 the vertical line segments open segments they are all open intervals in L so therefore they are all open they are all disjoint with each other and how many of them are there as x raise over 0 twiddle omega there is the uncountably many in a in a in a topological space you have uncountably many disjoint non-empty open sets at each point you can take a select a point that will give you an uncountable discrete set that is not possible okay so second second countability is violated so L is not second countable if we extend L by by one more line you see on omega omega close we don't have a line you put one more line here okay then it is not path connected namely if there is no need to put one more line you just allow omega omega twiddle cross 0 this capital omega cross 0 take L hat to be this L union this capital omega cross 0 okay one more point then the same electrical graphic ordering and all that okay then this will be a larger space than L and that is not path connected so that is what I am going to prove it namely there is no path from 0 twiddle cross 0 to omega cross 0 we know it inside 0 twiddle omega because that is there are too many discrete points here but this is this path is inside L there is no path this is what we want to say for suppose you have a path path means what continuous function from a closed interval you can assume 0 1 to L hat joining 0 twiddle cross 0 with omega cross 0 since every point in L hat in fact except the starting point namely 0 twiddle cross 0 that is a cut point in particular all the points x cross 0 where x raise over 0 twiddle open omega open they are all cut points if you have a path from one point to another point and there is a cut point we separate these two points that means and if you have path the path must go go through that one this is like just a intermediate value theorem right so here you see the intermediate value theorem here alright so it has to go through that one that means what you have uncountably many points discrete points inside 0 comma 1 that is not possible okay so I make it much more clearer here so these are cut points follows that the image of tau image of tau must be comparison contains the whole of L because all those points are there every point so they must be in the image of tau by that intermediate value theorem the subset singleton x cross 0 1 the open interval they are uncountably union of mutually disjoint open subsets okay and there are inverse images in 0 1 open under tau we will contradict the fact that this one is second countable 0 1 open second countable again you have produced infinitely many disjoint open subsets so this if you just add one more point you will give a path connected space whose closure is not path connected okay we have seen such thing by topology Sankar also that is not a great thing but this example also gives you that okay yeah what we observed above implies that the one point complication of L is not path connected in fact this if we add just one point here instead of 0 1 that will be again one point complication of this one that will be compact space okay so one point complication of a path connected space is not path connected okay that is the stronger conclusion than saying that closure of a path connected set need not path connected okay in particular no neighborhood of the point at infinity is homomorphic to an open interval I am talking about one point complication in this sense we cannot extend L beyond the far end 0 this omega cross 0 1 okay if you add one more point here there will be a problem this wall 0 1 we can extend but there already there is a problem however we can extend L at the initial point 0 to L cross 0 that is what we have done indeed this filer Euler notation L is the answer okay on this side I can extend it for the same reason as before it follows that L cannot be extended any further on the left hand side also the right hand side you cannot extend we have seen on the left hand far end you cannot extend it so this is completely saturated in that thing therefore this L is called the longest line so note that each point of L has a neighborhood homomorphic to an open interval okay and that is why it deserves to be called a line it is a longest line because you cannot have another longer than of course this will be not second countable either because even the subspace this curly L is not second countable okay so this will be these remarks and great results here we have proved this chapter comes to an end the some of the lessons which you have learnt here some of the results we will use it in the next chapter also this example L itself will be repeated you know in a quoted as the you know illustration of the hypothesis that we put namely second countability in the definition of manifolds later on thank you