 Okay, so what we did, we proved two lemmas yesterday. And there's a third lemma still, so I will write this lemma, which is useful. So let X be a topological space. We have a topological space, X, X is a topology. And we have a collection of open subsets of X. So let's see, written, okay? So C be a collection of open subsets of X with a following property, such that. So this is the property. Sometimes this is the definition of basis, if the topology is already given. Such that, so for each U and U, so this is the condition. For each U and U, sorry, for each open set U. So let me give a name here, X. So this is X with a topology T, okay? T is a topology, X is a topological space. In general, we don't write, but T is better to write. So it's X topology, open set X in T, obviously, no. And each point for each open set and each point X and U, they exist. There is C and C, so printed, and this is written, okay? As usual, there's C and C such that X is in C is contained in U. So this is condition, the conclusion is, then C is the basis for topology. It is a basis, since C is the basis for topology on X. And the topology generated by C is T, it's the origin of topology, okay? We get the topology with three words. And the topology generated. And the topology generated by C is T, it's a given topology, T, okay? So two parts. It is a basis and it generates a topology and this topology is a given topology. So it's this condition here. And this is interesting because in some books, you first define topology and then a basis of a already given topology, okay? And then this is the condition, okay? The basis for a topology. This is exactly then the definition of basis for topology T on X, okay? Which is given. Then this is the condition. Here it's a lemma, here it's not a definition, but then we say C is the basis in our sense and it generates the topology which we want, okay? Then C is the basis for topology on X and the topology is generated by C is T. This is not T prime, okay? So C is the basis which generates T. I will not give the details, it's very easy. But so the proof, there's two things, proof. The first is C is the basis for topology. And this is the basis for topology. For some topology, for topology on X. So here we have our definition, two conditions, okay? And the two conditions is that each point is in some basis element. So we have this condition here, okay? There is for each open set, so what is open is X, for example, okay? The whole space is open. So for each point X, in X, in the whole space, we find the basis element, which goes on. So this is part of this definition. That's the case, U is X, okay? Then we have the first condition of the A, I will not write this, okay? We should follow in some sense, okay? So the first condition is clear because X is open. And the second condition, what is the second condition? The second condition schematically is you have for a basis. You have two basis elements. I don't write the text here, okay? And you have a point in the intersection. Then you should find the third one. So this was called B1 and B2. And then we find the third one here. It contains a point and B3 is contained in the same. That's the second condition for the basis. The more important one, okay? However, these are elements now of C, okay? And these are open anyway, okay? These are open. And so the intersection is open. These are open. They are elements of C, but the important is is a collection of open subsets, okay? We have to take open subsets. So they are open anyway. So the intersection is open, okay? Given a point in this intersection, which is open, we find a basis element, an element of C, which contains a point, where is it? And it's contained in this open set, no? So also the second condition is true. So it is a basis, okay? And then the second part is that it is a basis. So this is used in exercises very often, okay? You have something and you want to see this as a basis for a topology. If the topology is given already, okay? And then you don't have to prove it as a basis. It follows from this already, okay? It's part of this. It's easy, it's trivial almost, but you don't have to prove, okay? In general, you have to prove that it generates what you want that it generates, okay? So the second part is if T prime is a topology, well, it's maybe too complicated. You generate it by C, which is a basis. So it generates a topology, no? Then T is equal to T prime, okay? Then T is equal to T prime. It's the same, okay? That's the second. These are two eases. This is also very easy if you want to have two inclusions, no? T prime, the topology generated by a basis, which is now C. This is a basis now for some topology. The topology, how do we find this topology? There's a definition, of course, no? But we can have a lemma which says we take arbitrary unions of basis elements. That's our topology, okay? The basis elements are open. That's here, okay? They are open anyway in all topology. So arbitrary unions are open, okay? In our topology. So if you take arbitrary unions of elements of C, we remain in our topology. This means that this is one of these inclusions. Which one? So we take arbitrary unions. We get T prime. This is open, so this is certainly T, okay? Arbitrary unions of open are open. And then we have to prove that any element of T is an arbitrary union. But we have this condition here, okay? It's a condition that each point, if you have an open set and a point, then we find an element of C which contains a point and is contained in the open set, okay? Then it's clear that it's union of, we get each point, okay? It's union of open sets, okay? So also this is very easy. Except if you have to write it, it's, you have to write something, okay? But this is in some sense the steps of the proof, okay? You should think about this, maybe. I don't want to write because sometimes, okay, anyway, this is the idea of how do you prove it, okay? The steps, the main steps of the proof of this, okay? And it's a very useful lemma here, which we will use very often, okay? Good, so. Then we have, before we do examples, where we can define sub-basis. So this is the third definition. So we have topology basis definition, and this is now sub-basis. So X is again a set now, okay? We start with a set, X is a set. A sub-basis, now we want to define sub-basis. For a topology, which you still don't know, on X is a collection. You can guess how we write this collection. So it's written S now, okay? It's a written T, it's topology, written B is the basis, written S. It's a collection S, so this is a written S. Is that okay, S? Well, that's how I write written S, okay? Collection S, what do you want to define? Sub-basis, collection S of subsets of X, of course. Of subsets of X, and we have almost no condition, one small condition, such that whose union equals X. Each element is contained in some of these. Whose union is X? So you have the first condition. Each X and X's element is in some each, so you see the notation, each X and X for each, for each. If you want to write it as a condition, for each X and X, there is, so you know you see this is S printed, okay? And this is S written such that X is in X. S, okay, so there's remain one condition, and this is a trivial condition, okay? There's a sub-basis, so you can take everything, almost, okay? If the union is not X, you just add X to what you have, and then, of course, this is trivial, okay? You take also X as one of the, then the union is everything. And now we want, of course, a topology, so what we do is, how do we go from sub-basis, so sub-basis? We have a sub-basis S, and then we want to go to a basis, okay? To a basis B, and what do we do? We take, find it, these should be open sets, no? What we want anywhere at the end is S contained in B, contained in T, okay? These are all open sets, no? Sub-basis is the smallest, and the basis is important. So we take finite intersections, all finite intersections of elements of the sub-basis, okay? And then we get the basis, so we have a condition, two conditions now, a basis. The first one corresponds, finite intersections means that we get S itself, okay? Intersections of one, okay? So we get of two or three, no? We'll find them. So S is in B, so this is the first condition of a basis, and the second condition of a basis, well, you take two basis elements, two finite intersections, and you take the intersection, what do you get? Again, a finite intersection. So in this case, we are in the nice situation that the intersection of two basis elements is again a basis element, okay? So the B3, you take just the whole intersection, okay? So the second condition is trivial, okay? This is not always for a basis, but in some cases, the intersection of basis elements is again a basis element, okay? And this is here the case for this, so. The intersection of two finite intersections is a finite intersection, no? That's the point. And then, so it's a basis, and then we have a topology, T. And what should I write here? Arbitrary units, no? Finely intersections, and that's the lemma. From a basis to the topology, we go by taking arbitrary units, okay? So arbitrary. And then we have this, of course, okay? And T is the topology generated, defined in this way, okay? T is the topology generated by the sub-basis S, okay? That's the, so T is the topology, and B is the basis generated, if you want also, but T is the topology generated by the sub-basis S. So how to define it, if you have to write, you take arbitrary unions or finite intersections. That's how it's constructed, okay? Arbitrary unions or finite intersections? Arbitrary unions or finite intersections? Yes, one second, slowly. So you want to say it follows from, yes, yes, that's the only condition. What is this intersection of empty family? Empty, yes. Let's forget, this is very formal, okay? If you want a basis here, the first condition of the basis is exactly this. So then we need it, okay? So they have the same problem for the basis, probably, or? I mean, so that's another question. Why is this condition important, okay? Why is it important that any, so if you have an arbitrary point X, we need, in any case in the topology, the whole space should be open, okay? And that means that each element is contained in an open set, should be called, if not in another, then in the whole space, in the whole, okay? So say again what you, I mean, it's very formal. This is not, but this is, let's say, this is a good condition, okay? Oh, of course you can play a game with conditions and relax a little bit and see what remains, okay? But for topology, if you have a point, you won't also have neighborhoods of a point, open sets which contains a point, okay? Otherwise, I don't know what you get, nothing interesting because it didn't occur in measurements, right? So the condition is, okay? You can take any? Yes. Yes. Yeah. But let's say to this, okay? Well, maybe we can discuss after lecture, but it's better to stay with the clear things, not to, of course it's a nice game to relax conditions and to see what remains. If it's useful, it's another thing, no. Okay, these are the conditions, okay? Three conditions for topology, two conditions for bases, one condition for sub-bases. Now we should discuss examples. This is more, well, it's not, I don't know if it's more interesting, but it's, oops. So we have a long list of examples now of topologies of bases and so on and so on. Some you may know and others maybe not, but okay. So what is the first example? So we have to talk about an important example and the book is the order topology. So let me start with the order topology. So X now is a set, but it's an order set. What means order? Let X be, and so there are various words. I forgot even the word in the book. So it's linear order, total order. So there are various words, so an order set. Let X be an order set. An order means for us a complete order, linear order, okay? So there are various words for this. It's not clear for me which is linearly ordered. Linearly or ordered, or a complete order. I'll say what it is, of course. Complete order, that's ordered. Because otherwise, it's called partial order, okay? If it's not this order, then it's only partially ordered. What does it mean? So this means that we have an order relation, which is this symbol, okay? And here we have to be careful now. This means smaller, okay? If you want smaller equals, so this is smaller. And if you write this, just this, this is smaller or equal. That's not the same, okay? Smaller or equal. Because for some other relation, this is inclusion, okay? Inclusion means strictly included or equal, okay? If you recall, this means both, okay? If you want that it's different, then so it's usually different, okay? Well anyway, so what means total order? If x, y are in x and they are different elements, then if you have two different elements, then we can compare. Then x is smaller than y, or y is smaller than x. So we can compare any two elements. That's the total order, okay? We can compare any two elements. That's what we want always. Otherwise we have to say partially order, okay? And then there's one, so also, this is also sort of strange. So x smaller than x does not hold, okay? In this order, okay? x smaller than x does not, is not true, okay? x is not smaller than x. It's equal to x, but not small. Does not hold. And then there's one important condition, which is the transitivity, no? If, so the important condition is this now. If x is smaller than y, and y is smaller than z, then x is smaller than z, okay? This is an important condition, which is, okay? This is an order set, completely or linearly ordered, or totally ordered, or whatever set. And now we want, so the example, of course, are the reals, okay? Example, this, the main example are the real numbers, no? And now in analysis, the standard topology, which we didn't define still, but what do you take for a basis for the standard topology on R? You take open intervals. That will be a basis. Not the topology, but the basis. Now this is the basis, which defines the standard topology. Open intervals, okay? And this is a basis. Because the intersection of two open intervals, what is it? It may be empty, or it's again an open interval. So in this case, again, the intersection of two basis elements is either empty or it's a basis element, okay? This is, I will write the definition later, but this is the standard topology, okay? And now we do the same here, okay? We have an arbitrary set, so we define the basis. So what do we do? We take open intervals, okay? So we take x, y, x smaller than y. So of course, x, y in x. So what is this? This is our notation for open interval. And these are all points z in x, such that of course it's between x and y. That is between x and y. That's an open interval, okay? And this is our notation for an open interval. So somebody always write is used to this and then they go on writing this, no? This has also some sense to write this because this is sometimes used for other notation, no? But in the book and we follow this for an open interval. So open interval. And now we have, already is a problem, okay? This is in general not a basis. We don't get something, because example, we can take here the interval, whatever, zero, one. This is ordered, okay? It's a subset of order is ordered, okay? So this is ordered. And now we have this as a set, no? And we take this and then we have a problem because then these two points are in no open interval in this set, okay? So it's not a basis, all this is what we see, okay? The first condition is not valid, okay? So we have to be careful, so union. So we have to add. In fact, for the real, this is sufficient, okay? But if there's the smallest element, then we have a problem and then we have to add something. So we take x zero, y, x zero smaller than y, if x has the smallest element, x zero, if it exists, okay, the smallest element. If it exists, it's unique, of course, no? Because we have a complete order, no? There's one, but maybe no. If x has the smallest element, we, so what is this? This is a half open interval, okay? Half open interval. So these are all points z such that z is more, bigger equal to x zero and smaller than y, okay? So here's, it includes this part, okay? So you know you see how to write intervals, more closed intervals. Of course, then it's clear that we have to add something still if there's a larger element. We have the same problem. So we take y, well let's take x, y zero, and here now, okay? If, so again, x smaller than y zero, if x has the largest element, y zero, okay? If x has the largest element, y zero, then we have to add these two. Then b is the basis, okay? So, then b is the basis. It's the basis, which generates, by definition, the order topology, okay? Associated to the given order, which generates, and this is called the order topology. Of course, associated to the given order, okay? Other order, other topology, associated, associated to the given order, of course. So this is the definition of order topology. It is the basis, no? Because every element is in some of these, okay? And then we need these two, if you have the largest or smallest element. And the intersection of two basis elements is empty, maybe? If it's not empty, it's, again, one of these three types. Intersection of two open intervals is an open interval, or empty. The intersection of this and this, well, you have, it's clear, you see the various cases now, and you always get an interval. Or it's empty, okay? But picture, if you want, no? You make a picture, you have the smallest element, and this, okay? And now you intersect with the same, if you intersect with the largest element, then this is an open, okay? That's one possibility, and so on, and so on, okay? So make a picture of all possibilities, okay? Good, so this is the basis, and this is, yes? No, no, no, no, no. This picture, you mean? No, I like to, if I have an ordered space, I make always this picture, because it's a linear, it might be R, yes, okay? It looks like it's a real, okay? But I make also for an arbitrary ordered space, no, X. Of course, it's not a continuum or something in general. That's clear, no? Because what is ordered is the natural numbers, no? All subsets, no, are ordered. So the natural numbers, the rational numbers, whatever, okay? But you see they come one after the other, okay? Maybe it's not everywhere, something I'm missing or something like that. But for an ordered set, I like to make this picture, okay? It's linear, it's linearly ordered, okay? Maybe some points are missing here or something like that, but it's in general for, it's just a picture, okay? And then I can interval, right? It's like for the reals, okay? I have this interval, I have these intervals, I have these intervals, and I have closed intervals, okay? It's the same for the reals, but it's for arbitrary linear order. Schematic picture. So this is the ordered topology. And then there's a strange example. No, let's see, the first, sorry, one second. Something, I define the ordered topology and I should define, okay, so here's a very strange example, but it will be important in a certain, so, so example. Of course, if you take the reals with the natural order, you get the standard topology, okay? If you take the natural, so, well, I take R2 now, R2, R cross R. So you don't see a natural order, okay? The plane has no order, okay? It's not a line, no? But we have an order, of course. So we take the, how is it called in the book? We take the, what? No, no, no, not so complicated. Well, it's not an analysis. We take the dictionary order or the lexicographic order. What is the name of the book? Dictionary, okay? We take the dictionary order. Or if you want lexicographic order, okay? So, it's like the lexicon, okay? You have two letters over here, okay? You have all words have first letter, second letter, okay? And you can compare as a lexicon. That's first, you look at the first, then each R has a natural order, okay? It reels if you don't say anything, it's a natural order. So what does it mean? Now, how to write elements? And this is another feature of the book. So I take two elements, X, and now you say you write, don't write them, okay? This would be an element here sometimes, okay? X, Y. But this is also an open interval. So we get some confusion here, okay? So in the book, you write elements of a product, not in this way, but in this way. X times Y, let X times Y, X1 times Y1, okay? And X2 times two, it should be two elements. X2 times Y2 be in R cross R. And that's the way how we write this, okay? Not as an interval, but otherwise, we have some confusion between the two notations. So then X1 cross Y1, so these are two words, smaller than X2 cross Y2 if, so it's a lexicon, okay? You look at the first letter. If X1 is smaller than X2, it comes first in the book, okay, in the lexicon, the first letter. Or they might be equal, no? And then you look at the second letter. And Y1 is smaller than Y2. If they are also equal, then they are equal anyway, okay? So this is a lexicographic order, or the dictionary order on R cross R. What topology do we get? We get, in some sense, a strange topology. So let's look, have a look at this topology. There will be one important example which comes from this topology. So the basis, what, open intervals now, okay? Because we have no smallest element, no largest element, no, in this case. Open intervals, no, no, no, no, no rectangles. One second, the order topology, okay? There are open intervals, or half open if you have the smallest or largest element, okay? We don't have the smallest, largest element here, clearly, because in R there's nothing, okay? And so there are open intervals, okay? Basis are open intervals. That's the definition, okay? These are the two we don't need because we don't have smallest element, we don't have large, so they are just open intervals. Open intervals now, we have X1 cross Y1. Now we have both notations, no? This is one of these elements. And then the second point is X2 cross Y2. This is an open interval, and these are the two elements, okay? So this must be smaller than this, okay? For an open interval. And there are two cases. So let's have a look, how do they look like? Okay, they have two cases. We may have X1 equal to X2, okay? So if X1 is equal to X2, then Y1 is smaller than Y2, because this is smaller than this, okay? So we have here, what is Y1, Y2, Y1, no? Okay, and so we have this point here. This is X1 cross Y1, and this is X1 cross Y2, which is X2, and so we get this, open interval. All points between these two points, okay? Which I indicate in this way. This is an open interval, okay? In R2, in R2, okay? On this line, okay? So this is an open interval if they are equal, X1 equal to X2. If they are different, if X1 is smaller than X2, so now we have a different case, X1, and it's smaller than X2, no? Then I don't care too much about Y1, Y2, they're not important, okay, for the order. So here's the one point X1 cross Y1 and Y2, I don't know. It might be wherever, okay, here, okay? X2 cross Y2, but it might be also here, okay? So what is the interval now between these two, okay? The interval is, so we take everything which is larger than this and smaller than this, okay? So we go on this coordinate, okay? So this point is not there, no? It's an open interval, but if we go here on this line, then it gets larger and larger, okay? The second coordinate, no? The first one remains. So we get everything here, which is above here, okay? But we did not arrive here, of course, no? Then we go to the left, to the right, okay, a little bit. For all these points here, all these lines, the complete lines here, no? They are, all these points are larger than this but still smaller than this one, okay? And how does it end? Well, it's clear that it ends in this way, okay? So you get this, then you get all the lines between and then you finish with this. This is the largest. If you go here, then it's out of the interval, okay? So it looks like this, okay? Strange set. It's not a rectangle, open rectangles are the standard topology of R2. This is another topology. This is open, no? This is a basis element. This is basis element. So this is open, now, okay? In particular, it's open, this one. And what you see is these suffices. You don't need the second one, okay? You can take only the first one. Because if you have these, you have these anyway. If you take, these are so, are sufficient, the first type of intervals. Because the second type is a union of the first type, no? Each point here, you easily get, of the first type. Each point here, you easily get something of the first type. Now each point here, you get something here. So for the topology, you need only these. You take up to a union, no? But you get these anyway, okay? So you have these also. So to generate the topology, intervals. So we have the smaller basis, okay? So we take just these, okay? And then you think, is this a basis? So each point is in some of these, yes, no problem. The intersection of two of these in general is empty, no? If it's not empty, it's one of these again, okay? So this is a basis, okay? So we have a smaller basis, okay? So to generate the topology, intervals of type. So x cross y one, x cross y two suffice. You can take these, this is already a basis. Okay, that's a strange example, but we will see this again, okay? So it's not just an example, it's in some sense important. Well, it's clear, okay? We have R2 with a strange order, not a strange order. R2 has no natural order in some sense, no? No geometric order, okay, like a line, okay? So we have the lexicographic order, the dictionary order, and this gives the topology in R2, okay? This is a dictionary order topology on R2. So this is an example, which we will see again. Okay, so what? So this was an order topology, and that was confused because, well, it doesn't matter. There's another example, which is important in the book, which is a lower limit topology on R. So we have many examples, as I said, no? And now we have two topologies, the standard topology, so we compare two topologies. The standard topology, and the lower limit topology, so this is limit topology, and here we take the real, okay? So we have the standard topology, and we have another topology, which is called lower limit topology on the reals. This is also an important example. I mean the standard topology is the most important topology, but so we take a basis, B, for the standard topology, as we said, we take open intervals, okay? So X, Y, so recall, this is an open interval again, no? Open interval, real interval, okay? We have the reals, X smaller than Y, X, Y in R, is a basis, which generates the standard topology by definition, that's our definition, that's the easiest way to define the standard. It's a basis for a topology, which generates the standard topology. And I emphasize, this is the easiest definition of the standard topology, okay? This defines the standard, by definition. This defines the standard topology. This is also the definition of the standard topology, for us, the definition of, maybe it's clear from analogies of the standard topology. That's the definition, okay? That's the standard topology. And the lower limit, no? The lower limit, you just change a little bit, so you take B prime. And now you said, well, lower limit, I take X, Y, I take not open intervals, but half open intervals, and again, X smaller than Y, X, Y behind the reals, on the reals. And now you think about this, and what you see, well, it's a basis again, no? The intersection of two of these is again of the same type, or empty, okay? So this is a basis, this is a basis also. The intersection of two of these is again of this type, no? If you don't think, I mean, I like to see geometrically, no, I make these pictures, no? No, it's the reals, but it's good for any space, okay? So this is a picture of half open interval, no? And you take two of these, and what you see is intersection is again of this, okay? It might, of course, be empty, no, somewhere here, or it might be include the other one, but in all cases, it's no problem. So it's a basis, and this is a lower limit, which generates, by definition, the lower limit topology on the reals. So this is lower limit topology on the reals, and this is standard topology. And then you have an observation, so can you compare these topologies? What would you say? What is the relation between these two topologies, the standard and the lower limit? Yeah, well, I mean, compare means finer, strictly finer, coarser, strictly coarser. So the lemma, if you want, then, no, no. The lower limit topology is strictly finer than the standard. The lower limit topology is strictly finer the standard topology. The lower limit topology on the reals, we are on the reals here, no? We are on the reals, and this lower limit topology on our strictly finer than the standard, one hour. But for other, we have no standard topology. Standard topology is for the reals for the moment. For R2, we still didn't define, okay? There's also a standard topology. So proof, the picture. So we have to prove two things, no? We have to prove finer, okay? So fine, okay? It's finer, and that's strictly fine. So finer, we have a lemma, okay? We have two bases, okay? And yesterday, the last lemma was a lemma which compares two topologies, okay? Finer, apply a lemma. Well, I don't give numbers here, but apply this lemma, last lemma, yesterday. And what does it say? So there was this condition, no? So let me, for each basis element, schematically, okay? Here on the reals, no, anyway, okay? So finally, these are the reals, I'm not some French other space. So for, we want to prove that lower limit topology is strictly, is finer than the standard topology. So we take an element of the standard topology, a basis element of the standard topology, okay? This is an open interval, standard topology, open interval. And we take a point, okay? Then, we should be able to find a half open interval, yes, exactly. And then the lemma says, then it's finer. What interval you take is clear, no? You take the interval, this is AB, no? Well, you take XB, no, that's a, so you take XB, okay? So X is in XB, and this is contained in AB, okay? And this implies finer. The lemma then says, it's fine, okay? This is a picture which says finer. Without what? Okay, sorry? Why did we prove the lemma yesterday? And the lemma is this condition? No, no, intuitively it's, that is finer, this? I mean, any open interval is a union of half open intervals. Because any point we find, no, it's okay, okay? Any open interval is a union of half open intervals, right? For each point in the open interval, you have a half open interval, no? So it's okay, right? Anyway, that's a proof, okay? This is a picture for a proof. And we proved the lemma yesterday, so. No, it shouldn't be strange, okay? You say half open intervals are too special or something. Open intervals are better. Well, if you have half open intervals in the basis, then you have all open intervals anyway, okay? Because they are unions of, these are. So now we have to put strictly finer, okay? And that's clear or not? So strictly finer, that's the second. Finer we have, no? But strictly, so what means strictly finer? It means there's one open set which is not open in the standard. But now you say any half open interval is not open in the standard, no? Clear, all right? So I take any, this is open in the lower limit, okay? This is open in the lower limit. It's a basis element, no? Half open interval, open in the lower limit. It's a basis element. In fact, it's a basis element, okay? A basis element, they are open. Basis elements are open. And this doesn't look open in the standard, no? Nobody would say that's not an open interval. There's an half open interval. But there's only one point where it goes wrong, okay? Now we apply again the lemma, okay? Which, the same lemma, we apply a lemma, okay? So we take AB, okay? The point A, of course, is the only problem, okay? There is no open interval. It's clear that A, there's no open interval which contains A, which is contained in AB, okay? Because in open interval, we have to go to both sides, okay? So there's no open interval containing and contained in, that's clear, no? That's trivial. And now the lemma says then the standard topology is not finer than the lower limit, okay? It cannot be finer than the lower limit. So they're not equal. So this is an example of an open set, open in the lower limit topology, a basal limit, but not open in the standard. Well, it's trivial, no? Standard is a topology from analysis. So we don't need even the lemma here for, well, anyway. Yes, the open interval from one to two, yes. Is that true or? Sorry, sorry. Yes, yes, these are basal limits of the lower limit, no? And I take the union over all n, so it comes clear, I'm actually close to one, okay? But we don't get one. No, no, we want to be larger than one plus, it's plus, yes, plus. That's true, right? So it's clear that this is open in the lower limit. It's a union of this kind of stuff, no? Right? Yeah, you can take one point five, no? For each point, you have half open, that's what we said here, all right? Yeah, but he says it's not, I mean, this is a proof, but it's still strange, no? And then he wants to see that one of two maybe, but it's this, okay? So it's, yeah, this was the other, okay, right? So it's strictly final, okay? The lower limit, RL. And this is an important example in the book. This is the first, there are four, five examples which are all over the book and this is the first. So RL, then we write RL, L, R with the lower limit. Okay, this is it, R with the lower limit. And this is one of the most important examples in the book, it's a strange space, but it's important. It's a lower limit to put. You will see it again and again. And R, if you write just this, no? This means R with the standard to put, okay? If you don't write anything and this is the problem with space, then it's a standard, okay? R with the standard to put it. So we don't give an index or something, it's just standard to put. Good, this is next example. So this we see, we see all everything again and again. So it's good to get used to order topology, to lower limit topology, because, but these are easy generalizations of, then you say, of course, lower limit. If you have lower limit topology, it's clear that we have upper limit topology, by the same argument, and the lower and upper limit topology, they are incomparable, yeah, that's true. They are incomparable. And it's just a matter of choice, okay? You choose one of these, and it's lower. As well, you might always take the upper, okay? So in the book, you don't find the upper, you find the lower. But the upper limit topology, then you would have this kind, okay? That's the upper limit topology, you know? But it's, for us, one is enough. So upper limit, so this would be something like upper limit topology, okay, bases. And this is not comparable to lower limit, okay? They are incomparable, not comparable. Lower limit, well, you see this by picture, no? You just take this one, this is what, lower limit, okay? And then, for this, this is a crucial point, no? And for this one, you don't find any element of the other type, which contains this point, is contained here. I mean, you can go to the other side, but you have to go here in some sense, okay? You go out, okay? So this is, and the same for the other, no? You have this for the upper, okay? And this is for the lower limit, this is a bad point. Because if you now want a lower limit, you have to take this, maybe this point itself, but you have to go here somewhere, okay, with the lower. So they are not comparable. This is a picture, okay? This shows that the, well, you need two pictures, no? Where here, lower, this? No, no. The other one? Yes. This? No. The other one? Yes. Yes. Which contains, yeah, yeah, right, right. So you want to give another proof of this in some sense, yeah. And then, of course, you may take, if something, but then you say, why not take also closed intervals? But this is not interesting, why not? If you take all closed intervals, it's a basis again, no? You get, now you can have smaller equal to y, also you get points, okay? The point is a closed interval, right? So what is the topology in this case? If you take this, all closed intervals, it's a basis. What is the topology? Points, discreet, yes. It's a discreet topology. So this generates discreet topology. So a is equal to a, a, no? Points are closed intervals, so this is nothing new, okay? This is not discreet topology. By the way, what is the, these are almost exercise now. What is the most efficient, so you have a topology and you want to find a good basis for the discreet topology, it's clear. What is the, for each topology as a basis, okay? What, you take all open sets. That's the basis also, okay? Not very efficient, no? You take everything, all open sets. That's the basis for the topology, okay? For the discreet topology, you have a smaller basis. What is this? Singleton, single points. That's the most efficient basis. Any basis has to contain single, of the discreet topology. Must contain single points, okay? Because they are open, okay? And this is sufficient to generate everything, no? Everything is open, so singletons, yeah. That's the most efficient basis for the discreet topology. Okay? In general, okay. So, other examples, I'm, it's, in some sense it's very often nicer to discuss examples, no, and to, but we have to go on with other examples. Important, more, maybe more important examples, which you know, more standard examples. So the next one is the product topology. So the product topology. On the product, X plus Y. So this, from the product topology, on X plus Y. So X and Y are now topological spaces already, okay? Topological spaces. And we want to define the topology on X plus Y. So, if you don't know this, you think, what, what, what is, well, you have, of course, R2, you know, from analysis again, R cross R, you know. The standard topology, you talk of open rectangles, you know, these are open, open rectangles are products, you know, of open intervals. So, that's what we, so this is a topology, this is open rectangles, it's not a topology, that's a basis already, you know. Because, okay, so we take as a basis, which is called, so the product topology. So we take B, we define it by a basis, you know, okay? So we take products, open times open. So U times V, U open in X and V open in Y. So in some sense, this is a basis, it's not a topology, it's a basis only, it's a basis, which generates, and that's the definition of the product, which generates the product topology. That's the definition of the product topology. On X cross Y, that's the definition. Why is it a basis? Well, schematically, each point, of course, is contained in some of these, X cross Y, what's the number, X cross Y, it's a, so the first condition is, as you would fulfill, the second condition is, you have two, the intersection of these. So it's a basis, so U one cross V one, intersection, the second one, U two cross V two. So now it's set theory, you know? What is this? U one intersection U two times V one intersection V two. So it's again the product, okay? So here we have an intersection of two basis elements, it's a basis. Here's the empty set, it's also empty times empty or something, no, empty, the empty set is here, so intersection of two basis elements is again an empty element. So we are in this situation. Every known segment happens. Doesn't happen all of us, no, because if you take open boards, then it doesn't happen, no. Intersection of two open boards is not an open board. It's a basis, so it's a basis. The picture by the way is this, no. So I like these schematic pictures, no. You have this and here is this. And you see this form, okay? Of course, you have U, this is U one, this is V one, and this is U two, and this is V two, no? And this is picture for this, what? Intersection is U one intersection U two times V one intersection U two, okay? So this is an illustration of this, okay? Of this set theoretic formula. It's better to see sometimes, okay? You make a picture and it's maybe it's not a question, it's U one, this is U one, this is U two, and this is U two. Now it looks somewhat better. Okay, so this is the definition of the product. Now, we have a lemma, which is useful, a small lemma. Suppose we have a basis for X and for Y, okay? So, by the way, in general, it's only a basis, not the topology, no? Already for the R two, okay? A union of open rectangle is not an open rectangle, no? Open sets and R two are very complicated, okay? You cannot describe directly, okay? You can describe these products, maybe. But even this, you cannot describe, maybe, okay? Because these are all open sets. So we take basis, so lemma. If I should use this notation, we have a basis for X, if I see it. If B, BX is the basis for X, let me write it this way, change the notation. BX is the basis for X, for the topology of X, for the given topology of X. And BY is the basis for Y, for the given topology of Y. Then, we don't have to take everything, so I have to take B prime now. This is B, okay? And now we have B prime. And what I take is not all products, but only for basis, of basis elements. So U prime times V prime. U prime times V prime, where? U prime is open, but it's in BX. And V prime is in BY. So we just take product of basis elements. It's a basis for the product topology. Oh, I have to go on here. It's a basis, it's a basis of the product of X cross Y. Well, maybe you can prove. So we are, I will not prove everything, but every now and then, so prove. So we apply the first lemma, okay? Apply a lemma. That's the first lemma today, you know? So what did this lemma say? We take, we have a collection of open sets. That was the first lemma today, you know? Collection of open sets. Such set, and the condition was given any open set and the point, we find an element of this collection which contains a point, it's contained, okay? This is this, any open set, any point, we find an element of this collection which will be the basis, which, so it's, so then let's do it, okay? Step by step, it's trivial then. So let, let W be open in X cross Y. And we need a point, no? Which is called X cross Y, X cross Y. By the definition of the product of Y, maybe you find this boring, but you have to be able to write this, okay? Without thinking too much, no? It should be almost automatic. By the definition of the product of Y, so unless you think what is the definition, no? The definition is something is open. If given a point, arbitrary point, you shouldn't find the basis element which contains a point, it's contained. Where the basis is the basis which defines, okay? So by the definition of the product, top of the X cross Y, there exists, open in X, no, well, sorry. Let me write it shorter. There is, so U cross V in, what was the name of the basis, B, which defines, okay? In B, some element here in B, such that our point, which is it, X cross Y is contained in U cross V, and this is, that's the definition of the topology generated by a basis, okay? This is the definition. And then it's a lemma that the topology generated is a union of all basis elements, but that's a lemma, okay? The definition was this. We change, of course, no? Take the other as a definition, and then you have this also, of course, okay? So it's clearly equivalent, such that. So what does it mean? X is in U, Y is in, but this is open in X, no? Because we are in B, open times open. So open in X, this is open in Y. By the definition of the basis B, no? But now we have a basis of X, right? Which is BX, so there is. So X is in U, U is open with a basis, so there is a U prime in B, the basis for X, such that X is in U prime, is contained in U, because this is the basis. And there is V prime in B, Y such that, where is it? Y is contained in V prime, is contained in V. Again, by the definition of the topology generated by a basis, no? The same definition also. So then, finally what we have then X cross Y, is contained in U prime, which is contained in U times V, which is contained in W. And this now is an element of B prime. And now the lemma says, so now this lemma, which we started with, oops, sorry. The lemma then says, no, the lemma which I wrote in the beginning and the proof is very easy, but it's very useful, no? The lemma says then B prime is a basis which generates the same topology, the product B prime, and that's the end of this proof, okay? Apply definitions. Example is, example, important example, of course. R2, R cross R, this is a product. The standard topology in R2 is the product topology, okay? The standard topology. That's a definition on R2, which is R cross R, it's a product topology. So each R has a standard, okay? The standard topology is a product topology. That's the possible definition of the standard topology in R2, the standard topology. So what is the basis? Yeah, now we have open rectangles. The lemma which we proved before, no? The preceding lemma says, now we have a basis here, a basis here which are open intervals, so the basis are open rectangles, okay? Now, a basis, it has a basis which is open product of open intervals, so x, y, x, one. Sometimes it's not so good to give. Basis are open rectangles which are products of open intervals, no? So we just have this, okay? Open interval times open interval, no? Didn't give names, so this is standard topology R. And this is a basis which is, this is a basis very easy to describe, open rectangles, okay? The topology is very complicated, no? Because open sets are extremely complicated in R2, no? I mean, you have almost everything you, I mean, this is R2, no? And now you take something which is extremely complicated, but open, what means open? This is open, open means you take any point here, wherever you take, you find the basis element which, an open rectangle which contains a point is contained in this, okay? So you cannot describe explicitly open sets in R2. You can, this basis is very simple, okay? Yes, products of two open intervals, no? So now we have also the standard topology in R2, no? Then R3, we can go on, okay, for finitely many. What else? Nothing else. Well, one example which might be into three minutes, so a strange example, it's an exercise almost. So we take R2 with the lower limit, with the, so in R2, we have the standard topology now, okay? But we have this other topology which is the dictionary order topology, okay? The strange topology, lexicographic or dictionary order topology, order, the dictionary order topology. By the way, now we can compare, the first exercise would be compare it with the standard topology, no? It's a, can we compare these two topologies, the dictionary order and the standard topology? So one has to think a little bit, no, we have only one minute, but yes, we can compare. And the answer is, it's strictly finer again than the standard, so it's strictly finer. Hopefully, yes, it's strictly finer than the standard topology. This is not a good sentence. R2 with the dictionary topology is strictly, the dictionary topology is strictly finer than the standard topology, than the standard topology. So the good, a good, better sentence, the dictionary topology order topology on R2 is strictly finer, that's better, from grammar, okay? So let's, let me one minute, this is an exercise, so we make the same kind of picture, no? You want to compare, so we have finer and strictly finer, no, finer. So finer, we have the basis for both, no? We have to take a basis element of the standard topology and we take the best basis, we can describe best, this is open rectangle, okay? So take an open rectangle, this is open rectangle, so it doesn't contain this border, no? Right, open rectangle. This is open rectangle. And now, given the point, of course, we find this vertical line here, no? And this is the basis for the lower limit. So this picture says finer, okay? For any point, it's clear that we can find this vertical line, okay? And then strictly finer again, okay? Strictly finer. Then we have to find just one open set, which is not open in the other one, okay? And of course, each of these intervals, no? That doesn't look open in the standard, no? You take any point and you have basis rectangles, open rectangles, if you take an open rectangle here, the only thing that you can do is something like this, no? But you cannot stay on the line, okay? So this is the picture for strictly finer, okay? It's not possible. This is for finer, okay? So they are comparable, the other one. The standard is the standard, the good topology, the other one is topology, which we will see anyway, okay? So we get used to it, okay? But the last question for today, sorry. So the dictionary order topology, we can describe in a different way, no? We have product topology. So the standard topology is a product topology, R cross R, okay? R2 is R cross R. All standard, okay? Yeah, it is everything standard. Now we have R2 with a dictionary. And now the last question for today, which is interesting, is this a product topology also? Can you, it's not the standard, no? R cross R, that's clear. But this is also a product, which one? Yes, exactly. Or better, discrete with R, okay? Yeah, exactly. So this is, in fact, R, exactly. Rd cross, and this is Rd, and this is a discrete topology. What is the basis for the discrete topology? Points, single points. What is the basis here, intervals? What is the product of a point with an interval? It's one of these guys, you know? This is a point times an interval, right? So we get exactly, so this is also, in some sense, the dictionary on R2 is nothing else than the product topology, R cross R, where the first R is a discrete, and the second has the standard, okay? That is another description of the, that's sort of nice, all right? So we have an example. But as I said, the one, the main example in the book is this one, RL, okay? We will see again and again. And the dictionary on R2 itself, but the certain subspace of this will be one of the main examples of the book. So this, we will see again and again also, okay? But I will describe next week.