taeto Posted April 30, 2020 Share Posted April 30, 2020 6 hours ago, Mordred said: This is an example of compactification for the topological spaces used in Calabi-Yau manifolds. Source being String Theory on Calabi-Yau manifolds by Brian R Green. That is how compactification is understood in mathematics. It is a topological concept. Unfortunately it has little to do with the use of the same term in physics, where it is an analytical concept, that is, it talks about sizes. E.g. the default compactification of \(\mathbb{R}^4\) is gotten by adding a single new element, which is usually written with the \(\infty\) symbol. In the context of the thread, this is not apparently what is meant. Link to comment Share on other sites More sharing options...

Mordred Posted April 30, 2020 Share Posted April 30, 2020 (edited) No it is still the same in physics. It isn't based on sizes but on the sets of differentials. In the Brian Greene example he isn't even referring to the sets of reals but the complex differential sets. The meaning of dimensional compactification doesn't change from mathematics to physics. Physics employs all mathematical rules and this is part of how many of the physics terminology is defined. Many charts used in parameter space are not volumes per se but can use the same coordinates on graph. Phase space is a good example where your graphing amplitude of a signal. A topological space doesn't require a metric space see the link above for examples involving sets. If you want to properly understand physics regardless of theory you must understand it's mathematics. Not the verbal explanations physicists use to describe those mathematics to make it understandable to the average reader. Edited April 30, 2020 by Mordred Link to comment Share on other sites More sharing options...

Mordred Posted April 30, 2020 Share Posted April 30, 2020 Let's give a simplified example. Let's take a metric space and assign 4d dimensions {x,y,z,t} now at each coordinate I wish to map how temperature varies over time. Now for extreme accuracy I want to measure at each infinitesimal coordinate so I can apply partial derivatives. (Calculus of variations) So we can assign a set to temperature at each infinitesimal coordinate. Now ordinarily that set (dimension recall this value can vary without changing any other value) would be infinite in possible range. However I can place a boundary at Planck temperature and 0 Kelvin. I have now a infinitisimal topological metric space. Whose region is defined by the boundary of infinitisimals (point like) and it's dimension has been compactified by the upper and lower temperature bounds. However as I am using infinitisimal regions I have another infinity problem in the number of spaces. So I must set some boundary to that. So as we can never measure below the Planck length I can now limit the number of infinitsimal spaces which limits the number of seperate temperature sets. Link to comment Share on other sites More sharing options...

taeto Posted April 30, 2020 Author Share Posted April 30, 2020 2 hours ago, Mordred said: Let's give a simplified example. Let's take a metric space and assign 4d dimensions {x,y,z,t} now at each coordinate I wish to map how temperature varies over time. Now for extreme accuracy I want to measure at each infinitesimal coordinate so I can apply partial derivatives. (Calculus of variations) So we can assign a set to temperature at each infinitesimal coordinate. Now ordinarily that set (dimension recall this value can vary without changing any other value) would be infinite in possible range. However I can place a boundary at Planck temperature and 0 Kelvin. I have now a infinitisimal topological metric space. Whose region is defined by the boundary of infinitisimals (point like) and it's dimension has been compactified by the upper and lower temperature bounds. However as I am using infinitisimal regions I have another infinity problem in the number of spaces. So I must set some boundary to that. So as we can never measure below the Planck length I can now limit the number of infinitsimal spaces which limits the number of seperate temperature sets. Thanks for giving more detail. I am already 99.9% familiar with differentials and 1-forms of manifolds. But your use of "compactified" still sounds different from the mathematical usage. In ordinary mathematical sense you would topologically "compactify" a space by adding more points and more open sets to obtain a compact space. You seem to be doing something else which has to do with a bounding (analytically) of coordinate ranges. Besides, adding "infinitesimals" usually means to add more points as well, in between the reals, so to speak. Link to comment Share on other sites More sharing options...

Mordred Posted April 30, 2020 Share Posted April 30, 2020 (edited) https://en.m.wikipedia.org/wiki/Compactification_(physics) See here the keep in mind I'm discussing dimensional compactification as opposed to flux compactification in the above link. Though in both cases given above in the link you are contacting a group or set to its finite portions to avoid infinities. Edited April 30, 2020 by Mordred Link to comment Share on other sites More sharing options...

MigL Posted April 30, 2020 Share Posted April 30, 2020 (edited) Although I am familiar enough with 'compactification ( in the Physical sense ), where dimensions are rolled up on Calabi-Yau spaces, to discuss it, I cannot do the math. But Taeto seems to have a point, as your own link Mordred, gives various ways to 'compactify' a space ( in the Mathematical sense ), which seem different from the use of boundary conditions to 'limit' a space. At the top of your link... "For the concept of compactification in mathematics, see compactification (mathematics)." which leads you to... https://en.m.wikipedia.org/wiki/Compactification_(mathematics) As I said, the two don't seem equivalent to me, but my math skills, especially topology, are wanting. Edited April 30, 2020 by MigL Link to comment Share on other sites More sharing options...

Mordred Posted April 30, 2020 Share Posted April 30, 2020 (edited) I fully agree there are numerous methods to compactify a group. I would well imagine there are methods I am not familiar with. It's nice to see a good discussion on the topic. It is a very commonly misunderstood term in physics particularly on forums. Some other methods is Aleksandrov compactification, Wilson Compactification, Hausdorff compactification and Stone Cech compactification. Edited April 30, 2020 by Mordred Link to comment Share on other sites More sharing options...

studiot Posted May 1, 2020 Share Posted May 1, 2020 16 hours ago, MigL said: Although I am familiar enough with 'compactification ( in the Physical sense ), where dimensions are rolled up on Calabi-Yau spaces, to discuss it, I cannot do the math. But Taeto seems to have a point, as your own link Mordred, gives various ways to 'compactify' a space ( in the Mathematical sense ), which seem different from the use of boundary conditions to 'limit' a space. At the top of your link... "For the concept of compactification in mathematics, see compactification (mathematics)." which leads you to... https://en.m.wikipedia.org/wiki/Compactification_(mathematics) As I said, the two don't seem equivalent to me, but my math skills, especially topology, are wanting. There is no difference between the mathematician's definition of 'compact' and the physicist's. Indeed this is explicitly stated at the beginning of your link. note the sentence which begins "the methods of compactification are various....." Quote In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space.^{[1]} A compact space is a space in which every open cover of the space contains a finite subcover. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape". 15 hours ago, Mordred said: I fully agree there are numerous methods to compactify a group. I would well imagine there are methods I am not familiar with. It's nice to see a good discussion on the topic. It is a very commonly misunderstood term in physics particularly on forums. Some other methods is Aleksandrov compactification, Wilson Compactification, Hausdorff compactification and Stone Cech compactification. Here you are talking about compactifying a group. Nothing wrong with that , not all sets are groups though mathematically the meaning is the same. But mathematically 'compact' and 'compactification' is about sets. Quote Duffey : Applied Group Theory for Physicists and Chemists Next consider the parameters of a given group to be plotted as cartesian coordinates in a Euclidian space. Such a space is called the manifold for the parameters. If all the elements are represented by points within a finite region of the manifold the group is said to be compact. But if there is no finite boundary in any direction the group is non compact. We like compact because it allows us to use general theorems like the Heine-Borel theorem as justification for the mathematics of our functions and operations on them eg calculus. We like compact surfaces and manifolds as they keep sets and their coverings under control. The same ideas are also used by Engineers, as this extract from "Introduction to Differential Geometry for Engineers by Doolin and Martin" shows. Note their comment about research papers! 2 Link to comment Share on other sites More sharing options...

MigL Posted May 1, 2020 Share Posted May 1, 2020 Thanks Studiot. That clears up things. Link to comment Share on other sites More sharing options...

Mordred Posted May 2, 2020 Share Posted May 2, 2020 (edited) 13 hours ago, studiot said: There is no difference between the mathematician's definition of 'compact' and the physicist's. Here you are talking about compactifying a group. Nothing wrong with that , not all sets are groups though mathematically the meaning is the same. Though I edited portions of your post I quoted the sections I like to highlight in full agreement. +1 Edited May 2, 2020 by Mordred Link to comment Share on other sites More sharing options...

taeto Posted May 2, 2020 Author Share Posted May 2, 2020 (edited) 22 hours ago, studiot said: There is no difference between the mathematician's definition of 'compact' and the physicist's. It is possible that I am mislead by the page https://en.wikipedia.org/wiki/Compactification_(physics) and its first line In physics, compactification means changing a theory with respect to one of its space-time dimensions. Instead of having a theory with this dimension being infinite, one changes the theory so that this dimension has a finite length, and may also be periodic. It suggests that being 'compact' in physics has something to do with being small. Which is more like the common use of the notion, as in 'a compact car'. Whereas in mathematics 'compact' has nothing at all to do with size. Topology has no concept of 'length' or 'volume'. The Euclidean real line \(\mathbb{R}\) and the open interval \( (0\, ; 1) \) are topologically the same and non-compact. Whereas \( \mathbb{R} \cup \{\pm \infty\} \) and \([0\, ; 1]\) are identical and compact. If the concepts of 'compact' agree between mathematics and physics, then compactification of a dimension could introduce a new coordinate with range \([0\, ; 1]\) but not \( (0\, ; 1)\). In contrast, the wikipedia page does not seem to think there is a difference. Adding something to say "this dimension has a finite length closed interval as its range", though clumsy, might make it clearer. And beware that the explanation in the textbook refers to Euclidean space. This is a special kind of space for which it is indeed true that 'compact' means the same as 'closed and bounded'. This is not generally the case. The example of the compact non-Euclidean space \( \mathbb{R} \cup \{\pm \infty\} \) already shows it. Edited May 2, 2020 by taeto Link to comment Share on other sites More sharing options...

studiot Posted May 2, 2020 Share Posted May 2, 2020 22 hours ago, studiot said: Here you are talking about compactifying a group. Nothing wrong with that , not all sets are groups though mathematically the meaning is the same. But mathematically 'compact' and 'compactification' is about sets. My apologies I realise that this was ambiguous. To explain the underlined text, I really meant that the meaning of compact and compactification is the same applied to various objects of mathematics and physics and is based on the definition in set theory. There is more than one way to view the subject and the formal algebraic statement about covers and covering is very concise and needs considerable expansion to tease out the important parts for compact and compactification, because not all covers or sets are suitable for the requirements. 5 minutes ago, taeto said: It is possible that I am mislead by the page Possibly. But I did quote the important opening paragraph from Wiki, which included 22 hours ago, studiot said: The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape". 16 minutes ago, taeto said: Topology has no concept of 'length' or 'volume'. Not quite Topology is sufficiently general that a metric is not required to form a topological space. But there is a thriving topolgy of metric topological spaces. The classic book on this is Sutherland's "Metric and Topological Spaces " - Oxford University Press. 19 minutes ago, taeto said: And beware that the explanation in the textbook refers to Euclidean space. This is a special kind of space for which it is indeed true that 'compact' means the same as 'closed and bounded'. This is not generally the case. The example of the compact non-Euclidean space \( \mathbb{R} \cup \{\infty\} \) already shows it. Yes indeed it is limited - I said it was an Engineering textbook. Engineers and Physicists need the more restrictive mathematics that guarantees the functions they are dealing with are sufficiently 'well behaved' for their purposes. That is the motivation behind their interest in this. A simple example would be in geometry we have general polygons, regular polygons and convex polygons. Dimensional theory in Physics requires the convex restriction. Link to comment Share on other sites More sharing options...

taeto Posted May 2, 2020 Author Share Posted May 2, 2020 48 minutes ago, studiot said: I really meant that the meaning of compact and compactification is the same applied to various objects of mathematics and physics and is based on the definition in set theory. Certainly this description is not entirely correct. Suppose you give a first lecture on Kaluza-Klein Theory, and you explain to the students that 'compact' and 'compactification' mean exactly the same in physics as in mathematics. Then you give them as homework to compactify one of the dimensions \(x\) of a 5-dimensional manifold, where \(x\) initially has range \(\mathbb{R}.\) The obvious way to compactify in mathematics is to add either \(\infty\) or both of \(\pm \infty\), as well as enough new open sets to recover a topological space over the new set. So how do you mark such answers, which are correct according to your lecture? Full marks, I suppose. But how do you then explain that this does not work in physics, other than explaining that compactification is indeed different? Link to comment Share on other sites More sharing options...

studiot Posted May 3, 2020 Share Posted May 3, 2020 On 5/2/2020 at 12:49 PM, taeto said: The obvious way to compactify in mathematics is to add either ∞ or both of ±∞ , as well as enough new open sets to recover a topological space over the new set. So how do you mark such answers, which are correct according to your lecture? Full marks, I suppose. But how do you then explain that this does not work in physics, other than explaining that compactification is indeed different? Just as there are many ways to define compact in Mathematics there are many ways to compactify. It is also true that Physics uses a different definition of metric from Mathematics, particularly evident in Relativity, both GR and SR. As to compactness in five dimensions, Einstein-Bergmann-barmann assumed that the fifth dimension is compact. I am not sure if this was not also the case with Kaluza? Perhaps you have more information? Physicists and Engineers like compact spaces because it gives them justification for much of their working. On 5/1/2020 at 1:05 PM, studiot said: Note their comment about research papers! Link to comment Share on other sites More sharing options...

taeto Posted May 3, 2020 Author Share Posted May 3, 2020 15 minutes ago, studiot said: Just as there are many ways to define compact in Mathematics I really do not know what you refer to here. You could say that if we restrict to Euclidean spaces, then we can define 'compact' to mean the same as 'closed and bounded'. But that does not change the meaning of the notion. It just means that we are applying a theorem which expresses that being compact is equivalent to being closed and compact in that special case of a topological space, where we have additional information about its structure. What I mean is that to say that we can explain the property in a different way in a special setting is not the same as saying that the property itself has gotten a different definition. What better examples do you have in mind? 30 minutes ago, studiot said: It is also true that Physics uses a different definition of metric from Mathematics, particularly evident in Relativity, both GR and SR. Mathematics defines 'metric' and 'metric tensor'. They are different things. What is a definition of 'metric' in physics that is different from the definition in mathematics, and which is not the definition of 'metric tensor'? 39 minutes ago, studiot said: As to compactness in five dimensions, Einstein-Bergmann-barmann assumed that the fifth dimension is compact. I am not sure if this was not also the case with Kaluza? Perhaps you have more information? Einstein, Bargmann and Bergmann knew about Kaluza-Klein already, and they tried to get a hold on electromagnetism using this same idea, without success. The point is that before you can reasonably 'assume' a coordinate system based on a space in which some or all of coordinate ranges are finite in size, then it is best to know about the existence of such spaces, and how to construct them. And the method is called 'compactification of dimensions', starting from spaces in which the ranges of the coordinates are not already compact. It is clear that 'compactification of a dimension' in physics does not mean the same as 'compactification' of the range of a coordinate in mathematics. Link to comment Share on other sites More sharing options...

taeto Posted May 4, 2020 Author Share Posted May 4, 2020 Suggestion: I will edit the beginning of the wikipedia entry on compactification_(physics) which now reads In physics, compactification means changing a theory with respect to one of its space-time dimensions. Instead of having a theory with this dimension being infinite, one changes the theory so that this dimension has a finite length, and may also be periodic. into In physics, compactification means changing a model of [[spacetime]] with respect to one of its coordinates. Instead of having this coordinate being infinite, one changes the model so that the coordinate has a [[compact space|compact]] range of finite length, and it may also be [[Periodic function|periodic]]. Motivation: The entry in wikipedia for 'dimension' says that this is the number of coordinates used to describe a space. It doesn't seem correct to say that the dimension of spacetime is infinite. It is a coordinate (function) that can be infinite. Changing a coordinate representation of spacetime does not 'change a theory', it changes a model. Requiring the compactified coordinate be compact is the issue I want to ask the experts about. I think that I get from Mordred's contributions that this is a correct interpretation. And if so, then I would like the wikipedia entry to express it. The existing phrase implicitly indicates that it does not have to be compact. But how can I be certain that it isn't correct, and there are instances of 'compactification' in physics which modify a coordinate into a 'small' but non-compact function? It appears that Kaluza-Klein and Yang-Mills do insist on a compact outcome. Other improvements will be much appreciated. Link to comment Share on other sites More sharing options...

Mordred Posted May 4, 2020 Share Posted May 4, 2020 (edited) The above could be considered incorrect. A topological space can a set of points with a line through it. However this example has no distance function. A metric space must have a distance function. A metric spaces are topological spaces but not all topological spaces are metric spaces. The term spacetime itself is erroneous one can topological spaces in Euclidean etc where there is no need for a time dimension or another example being a two dimensional phase space. A topological space used to be defined by four Hausdorff axioms. Whether that's still true or not I wouldn't know. Edited May 4, 2020 by Mordred Link to comment Share on other sites More sharing options...

taeto Posted May 4, 2020 Author Share Posted May 4, 2020 (edited) 32 minutes ago, Mordred said: The above could be considered incorrect. A topological space can a set of points with a line through it. However this example has no distance function. A metric space must have a distance function. A metric spaces are topological spaces but not all topological spaces are metric spaces. The term spacetime itself is erroneous one can topological spaces in Euclidean etc where there is no need for a time dimension or another example being a two dimensional phase space. Indeed. The entry seems to want to describe 'compactification of a dimension'. This expression implicitly says that dimensions, coordinates, and a metric are all present. I will revise accordingly. A topological space is not 'a set of points'. It is a pair of sets, one of which is a set of 'points', the other is a set of 'open subsets' of the set of points. Compactness is not a property of the set of points. E.g. the interval \([0,1]\) is compact in the usual Euclidean topology induced by Euclidean metric. It is non-compact in the discrete topology, in which every subset is open, because you can cover it by singleton sets, in which case you need to have continuum many of them. You can compactify this particular space by adding one new element \(\infty\) and one new open subset \([0,1]\cup \{\infty\}.\) Yes, a metric space has a natural topology induced by the metric, in which the 'open balls around points' determine a basis for the topology. If we talk about a metric space, then we can naturally talk about topological properties such as 'compactness', 'connectedness' etc. in terms of that topology. We remain aware that other topologies on the same set are possible, such as the discrete topology and the trivial topology (in which the whole set and the empty set are the only open sets), and those topologies exhibit different properties. Spacetime spilled over from the original entry. I am not sure exactly how to deal with it. But it seems the primary example of the use of 'compactification of a dimension' to appear in physics. A phase space is another example, so I should include it. Hence: In physics, compactification of a dimension means changing a model of [[spacetime]] or a [[phase space]] with respect to one of its coordinates. Instead of having this coordinate being infinite, one changes the model so that the coordinate has a [[compact space|compact]] range of finite length, and it may also be [[Periodic function|periodic]]. Edited May 4, 2020 by taeto Link to comment Share on other sites More sharing options...

studiot Posted May 4, 2020 Share Posted May 4, 2020 (edited) What about relatively compact? Also what about the difference between a metric space and a normed space ? Normed spaces can also be compact. Edited May 4, 2020 by studiot Link to comment Share on other sites More sharing options...

Mordred Posted May 4, 2020 Share Posted May 4, 2020 (edited) If want a more precise definition I would suggest adopting how Wolfram defines topological space. https://mathworld.wolfram.com/TopologicalSpace.html PS the points I gave was an example not a definition. A mathematical definition must be exact including the applicable axioms and lemmas. Edited May 4, 2020 by Mordred Link to comment Share on other sites More sharing options...

studiot Posted May 4, 2020 Share Posted May 4, 2020 (edited) The subjects of covering, compactness, dimension and related topics are still under study and development in Mathematics. This thread seems to have picked on dimension so it would surely be appropriate to define a dimension. For the most part the old fashioned definition :- "A configuration is said to be n-dimensional if the least number of real parameters needed to describe its points is n." However Cantor and Peano developed counter examples to this definition since it is possible to put the points of a line into 1:1 correspondence with the points of a plane or cube. If we wish to base the discussion on set theory a good place to start is Menger's axioms D1 : The empty set has dimension -1 D2 : The dimension of a space is the least integer, n , for which every point has arbitrarily small neighbourhoods whose boundaries have dimension less than n. Edited May 4, 2020 by studiot Link to comment Share on other sites More sharing options...

taeto Posted May 4, 2020 Author Share Posted May 4, 2020 (edited) 3 hours ago, Mordred said: https://mathworld.wolfram.com/TopologicalSpace.html The description there is indeed similar to the standard definition of a topological space. It does contain the unfortunate phrase 'arbitrary number', when it should have read 'arbitrary collection'. Sometimes 'number' is understood to be a natural number, and that is certainly not the intention. 2 hours ago, studiot said: "A configuration is said to be n-dimensional if the least number of real parameters needed to describe its points is n." It is true that we say that. But I suspect it is because we know from linear algebra, or similar, that this number n is an invariant of the configuration. For the example of a well-known space such as \(\mathbb{R}^2\) we are used to familiar coordinates \(x\) and \(y\) to communicate the position of points. But we are still free to choose a \(y\)-axis at 45 degrees angle to the \(x\)-axis and transform it by \(y \to -y^3\) if we prefer. It does not change the basic properties of the structure in any way. It still has points with two coordinates \((x,y),\) only they are in different places than what we are used to seeing. So even for such a familiar 2-dimensional object it is not like we can freely speak of some particular dimension, unless we know how the structure is coordinatized. Edited May 4, 2020 by taeto Link to comment Share on other sites More sharing options...

studiot Posted May 4, 2020 Share Posted May 4, 2020 (edited) 13 minutes ago, taeto said: 2 hours ago, studiot said: "A configuration is said to be n-dimensional if the least number of real parameters needed to describe its points is n." It is true that we say that. But I suspect it is because we know from linear algebra, or similar, that this number n is an invariant of the configuration. For the example of a well-known space such as R2 we are used to familiar coordinates x and y to communicate the position of points. But we are still free to choose a y -axis at 45 degrees angle to the x -axis and transform it by y→−y3 if we prefer. It does not change the basic properties of the structure in any way. It still has points with two coordinates (x,y), only they are in different places than what we are used to seeing. So even for such a familiar 2-dimensional object it is not like we can freely speak of some particular dimension, unless we know how the structure is coordinatized. You entirely missed the point of my comment. Is the dimension of R^{2} two or is it one since every point in the plane can be referenced from a suitably winding line. Edit and what about relatively compact and normed spaces? The subject is much more complicated that at first sight when you delve deeper. Edited May 4, 2020 by studiot Link to comment Share on other sites More sharing options...

taeto Posted May 5, 2020 Author Share Posted May 5, 2020 16 hours ago, studiot said: You entirely missed the point of my comment. Is the dimension of R^{2} two or is it one since every point in the plane can be referenced from a suitably winding line. Possibly. Viewed as having only the structure of a set you could in principle coordinatize \(\mathbb{R}^2\) using a bijection to \(\mathbb{R}.\) Despite that, I suspect at least two separate reasons for not declaring the 'set-wise' dimension equal to one. For one thing we have the superior concept of cardinality to make a much finer distinction between sizes of sets. And second, we prefer to restrict the concept of dimension initially to vector spaces, and then extending it also to manifolds. These structures allow coordinatizations that are faithful to their respective topologies, in the sense that any two points that are sufficiently close together in the space have representations that are also close together. That would not be the case for your coordinatization of \(\mathbb{R}^2,\) as the coordinates of the points in an open set would be scattered all over the real line. Actually the concept of dimension gets applied to a structure like a finite graph in a different way than by looking at its graph theoretic properties. Instead its dimension gets defined by associating a vector space with the graph and then identifying the dimension of the vector space with the dimension of the graph itself. 16 hours ago, studiot said: and what about relatively compact and normed spaces? From which point of view? They describe special cases of compact sets and vector spaces, respectively. Link to comment Share on other sites More sharing options...

studiot Posted May 5, 2020 Share Posted May 5, 2020 (edited) 58 minutes ago, taeto said: From which point of view? They describe special cases of compact sets and vector spaces, respectively. What did one Mathematician say to another that made him blush? Spoiler My generalisation is more general than yours. Special cases, by definition, must be incuded in the most general case. 58 minutes ago, taeto said: And second, we prefer to restrict the concept of dimension initially to vector spaces, and then extending it also to manifolds. These structures allow coordinatizations that are faithful to their respective topologies, in the sense that any two points that are sufficiently close together in the space have representations that are also close together. That would not be the case for your coordinatization of R2, as the coordinates of the points in an open set would be scattered all over the real line. I don't prefer. Using the word 'restrict' automatically invoke the 'special case'. Yes all you say is true about coordinatisation and respective tolopogies. But coordinates frames are not the only aspect of topology, connectivity is another. Birkhoff and MacLane for instance display an example of generalising the concepts to sets without coordinate systems, pages 371 - 373. Fascinating that you can apply a cover to a lattice or even more general sets. Edited May 5, 2020 by studiot Link to comment Share on other sites More sharing options...

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