Tips for reading Grothendieck's EGA/SGA/FGA trilogy

[mathwonk]

I now have a little more experience with reading Grothendieck's EGA, although very little. I have been trying now for several years to read every word of Mumford's redbook, and finding it very rewarding but also challenging. I got especially bogged down lately in chapter II.8, "specialization", which is very algebraic, and quit cold for a while, but I am making progress again. What I have noticed is something that others have said here, namely it appears that in a way Mumford's redbook, perhaps like Hartshorne, is in a sense harder to read than EGA, because Mumford leaves out a lot of details, while Grothendieck and Dieudonne' include almost every possible detail in EGA. So for example in chapter II.8 of the redbook, I have encountered several statements and implications, for which I have had to provide fuller statements myself, as well as some proofs. For most of these I have been able to fill in the gaps, partly by reading up the needed algebra in Zariski and Samuel, but in at least one case, I found that Grohendick's EGA actually has a full explanation of what I was guessing at from Mumford. So in a sense EGA is "easier" to read by giving all the details, but at the same time, it seems to me much harder to read because it is so long. I.e. if you can fill a detail in Mumford by yourself, then you are better off doing so, rather than slogging through all that stuff in EGA. I.e. Mumford's book is less than 225 pages, while EGA has 8 volumes, the first alone being 227 pages, (all 8 total 1800 pages). So maybe one could profitably read Mumford and then refer to EGA for just those details one is stumped by, unless one really has the time and patience to plod through all of EGA. Note however that EGA I begins with a "chapter zero" with 79 pages just of algebraic background, before the definition of an affine scheme, and there are corresponding extensions of chapter zero in other volumes. Mumford on the other hand has a lovely first section called "Some algebra", which is only 4 pages long, and yet gives you the main results needed.

Nonetheless, given that our OP actually seems to enjoy what I would call gratuitously abstruse treatments of the subject, he may be happiest just plowing into EGA, as I myself tried to do some 50 years ago, right from the beginning. Of course in my case the result was complete discouragement and complete abandonment of the project all these decades, at least until now. So again I would suggest to most beginners to get some feel and intuition for algebraic geometry from Shafarevich and/or Fulton's or Rick Miranda's books on curves/ Riemann surfaces, and then trying Mumford's redbook, while relying on EGA as a sort of dictionary or encyclopedia for backup. Oh yes, and now I realize that Mumford-Oda, the expansion of the redbook (475 pages), may also be useful as filler of details in the redbook, and is available online I believe. It is also for sale from the Hindustan Book agency, and I just treated myself to a hardbound copy, having for years had a loose leaf (unfinished) copy given to me by Mumford some 40 years ago.

Another suggestion for why reading Mumford with EGA as backup is useful, I think you may get a better idea of what is important rather than just what are all the details, i.e. you may be able to better see the forest as well as the trees and the leaves. So I suggest you decide what is your goal, to understand the overall structure of the subject, with some details possibly missing, or to bask in some particular details thoroughly, possibly without grasping where they fit into the big picture.

 

[MathematicalPhysicist]

@mathwonk I prefer a book that covers every detail, even if it means more than 1000 pages to read than a slim book that doesn't.
Especially if those "gaps" in those books are hard to complete by yourself!

 

[MathematicalPhysicist]

I mean eventually, the pages of "gaps" that are meant to be completed by the reader are those missing pages that Mumford is leaping over.

 

[mathwonk]

I appreciate that, and that is why I tried to make clear what I see as the differences between the various books. It is so hard to recommend "best" books for people because we all have different criteria for what is a good book for us. I am realizing now that Grothendieck's contributions seem to have started out, at least in print, as a very succinct collection of seminar talks, mostly just announcing his results. This short summary version still is available in a volume called FGA. Then various people began the job of filling in his details, first in seminars at Harvard in the 1960's, others in Bourbaki seminars, with the most detailed version being EGA, possibly actually written by Dieudonne'. Even if you like this most detailed version best, a serious shortcoming it has is that it was so detailed that it never got finished! So many of the results described briefly in FGA never appeared at all in EGA. E.g. if you want to learn Grothendieck's theory of the Picard scheme, you are forced to get it from one of the other possibly less detailed accounts. (There is now a book in English titled FAG, or FGA explained.) And if you want to learn his generalized Riemann Roch theorem you must consult either his seminars or the presumably more detailed paper by Borel and Serre (http://www.numdam.org/article/BSMF_1958__86__97_0.pdf)
or the book by Fulton, or maybe other accounts.

Since life is finite you still might want to peruse either FGA, FAG, or even more brief, Grothendieck's talk at the International Congress of Mathematicians, just to see what he did, and then consult various longer versions for however many details you need. Although you say you like to see full details, for most of us it seems a frustrating but true fact that the more details we supply ourselves, the better we actually understand them.

I suspect that the people writing these more detailed explanations of Grothendieck's works are doing so in order to understand it themselves, since writing out the details is the best way to accomplish that. (There are such current projects at Columbia, the stacks project, https://stacks.math.columbia.edu
and at Stanford, the course notes of Ravi Vakil, http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf
which you might find useful). Indeed as I taught algebraic geometry just from the relatively elementary book of Shafarevich, I found many students, and I myself, had difficulty filling all the details he omitted, so I wrote out more detailed versions of his arguments and handed them out to my students. It frustrated me that one of my students declined to read my details but struggled himself with the arguments from less detailed accounts, until I realized to my chagrin that he was learning the material better than the others who were reading my details. In essence he who does the work learns the subject. But be your own judge, and best wishes and good luck and godspeed.

 

[mathwonk]

I have noticed that FGA already appears in the title of the original question, and that it was explicitly asked how to read the saga EGA/SGA/FGA. I am not qualified to answer this in full but I now have some impression which may be useful. It seems that FGA is a short account of most of the main results, without details, and that the other works were attempts to fill those voluminous details. The SGA series is apparently a sequence of Bourbaki seminars in which particular topics were addressed, and then finally EGA was intended to be the ultimate, most detailed version of everything. As stated above, for this overly ambitious reason, EGA does not actually exist in the full version that was contemplated. In particular the deepest topics seem not to occur there.

Thus of necessity one is obliged to struggle with the less detailed versions as well. An even shorter summary occurs in Grothendieck's address to the ICM in 1958, on the cohomology theory of algebraic varieties. Hence one possible approach is to read that talk, to find out what problems he was trying to address with his theory of schemes and cohomology, then peruse FGA, and then possibly choose some particular topics of interest to oneself to pursue in detail in one of the SGA seminars, such as the theorem of Riemann Roch, or Lefschetz's theorems. Or one could plunge into EGA for an elaborate development of the basic language of schemes, (without perhaps ever encountering some of the main applications that they were created for, I am not sure).

However, in reading Grothendieck's ICM talk, one sees that essentially everything he did was inspired by the seminal 1955 paper in the Annals of Mathematics, by Serre, universally known as FAC, or Faisceaux alge'brique cohe'rents, (Coherent algebraic sheaves), which was published originally in French. Hence it may be a good idea to begin by reading Serre's paper, and indeed I intend to do so when I finish Mumford's redbook. In fact my thesis advisor recommended these two sources as fundamental background in algebraic geometry many years ago. (There is even an English translation of FAC on the web.) Another advantage to this is that Serre's paper, like all his works, is very detailed and clearly written. Moreover in following Grothendieck's own footsteps, one has some idea of what he took for granted, or understood, when he planned and developed his own theory.

As a short hint to what one will find in the ICM talk, Grothendieck explained that among his main motivations were the desires to:

1) find a cohomology theory powerful and flexible enough to allow one to prove the famous Weil conjectures in arithmetic geometry. (There is a nice discussion of these conjectures in Appendix C of Hartshorne's book, where he says that the seminars SGA 4,5,7 were elaborations of one of Grothendieck's theories of cohomology intended to attack them.)

2) Find a general version of the Riemann Roch theorem, even one suitable for varieties that are not non singular, including a suitable theory of intersections.

If one is interested in the specific topic 2), there is the very nice paper of Borel and Serre, on the general Grothendieck RRT in the non singular case, linked above, and also the amazing and comprehensive book of Fulton on intersection theory, which treats the singular case and many other things. (The one 4 - star review of Fulton's book on Amazon seems to be evaluating the ease of reading of the book rather than its astonishing content and achievements. I also find it challengingly abstract to read, but we are lucky it exists.)

The Weil conjectures of course were solved by Deligne, and written up in some works in the elite journal of the IHES, but I am not familiar with them. I have only read a beautiful exposition of their content in an article by Dieudonne', in perhaps the "mathematical intelligencer", for which I have no reference. Wait, try this: (I love the subheading:"Daydreaming about algebraic topology".)
https://books.google.com/books?id=P...tical intelligencer, weil conjectures&f=false

 

[WWGD]

I have noticed that FGA already appears in the title of the original question, and that it was explicitly asked how to read the saga EGA/SGA/FGA. I am not qualified to answer this in full but I now have some impression which may be useful. It seems that FGA is a short account of most of the main results, without details, and that the other works were attempts to fill those voluminous details. The SGA series is apparently a sequence of Bourbaki seminars in which particular topics were addressed, and then finally EGA was intended to be the ultimate, most detailed version of everything. As stated above, for this overly ambitious reason, EGA does not actually exist in the full version that was contemplated. In particular the deepest topics seem not to occur there.

Thus of necessity one is obliged to struggle with the less detailed versions as well. An even shorter summary occurs in Grothendieck's address to the ICM in 1958, on the cohomology theory of algebraic varieties. Hence one possible approach is to read that talk, to find out what problems he was trying to address with his theory of schemes and cohomology, then peruse FGA, and then possibly choose some particular topics of inerest to oneself to pursue in detail in one of the SGA seminars, such as the theorem of Riemann Roch, or Lefschetz's theorems. Or one could plunge into EGA for an elaborate development of the basic language of schemes, (without perhaps ever encountering some of the main applications that they were created for, I am not sure).

However, in reading Grothendieck's ICM talk, one sees that essentially everything he did was inspired by the seminal 1955 paper in the Annals of Mathematics, by Serre, universally known as FAC, or Faisceaux algebrique coherence, (Coherent algebraic sheaves). Hence it may be a good idea to begin by reading Serre's paper, and indeed I intend to do so when I finish Mumford's redbook. In fact my thesis advisor recommended these two sources as fundamental background in algebraic geometry many years ago. (There is even an English translation on the web.) Another advantage to this is that Serre's paper, like all his works, is very detailed and clearly written. Moreover in following Grothendieck's own footsteps, one has some idea of what he took for granted, or understood, when he planned and developed his own theory.

As a short hint to what one will find in the ICM talk, Grothendieck explained that among his main motivations were the desires to:

1) find a cohomology theory powerful and flexible enough to allow one to prove the famous Weil conjectures in arithmetic geometry. (There is a nice discussion of these conjectures in Appendix C of Hartshorne's book, where he says that the seminars SGA 4,5,7 were elaborations of one of Grothendieck's theories of cohomology intended to attack them.)

2) Find a general version of the Riemann Roch theorem, even one suitable for varieties that are not non singular, including a suitable theory of intersections.

If one is interested in the specific topic 2), there is the very nice paper of Borel and Serre, on the general Grothendieck RRT, linked above, and also the amazing and comprehensive book of Fulton on intersection theory, which treats the singular case and many other things. (The one 4 - star review of Fulton's book on Amazon seems to be evaluating the ease of reading of the book rather than its astonishing content and achievements. I also find it challengingly abstract to read, but we are lucky it exists.)

The Weil conjectures of course were solved by Deligne, and written up in some works in the elite journal of the IHES, but I am not familiar with them. I have only read a beautiful exposition of their content in an aticle by Dieudonne', in perhaps the "mathematical intelligencer", for which I have no reference.

I am a bit confused on applying cohomology to varieties. I am by no means an expert in this area but I though varieties were too "rough" for cohomology, not being smooth or allowing for "standard" ( i.e. as in diff geometry) tangent spaces and that they would be better-suited to techniques of geometric measure theory , which deals with these " rougher" spaces.

 

[mathwonk]

all classical algebraic varieties over the complex numbers are triangulable, i.e. representable as simplicial complexes, even highly singular ones, hence ideally suited to simplicial (co)homology as developed by Poincare and others. So classical varieties are not at all too rough for ordinary topological cohomology theory. Indeed the singular (co)homology in topology makes sense for any topological space, no matter how rough. Of course we are speaking here primarily of sheaf cohomology, which is also quite general. The problem in abstract algebraic geometry is the extreme coarseness of the Zariski topology, and the fact one wants to work with varieties defined over say the integers. E.g. one wants to define a cohomology theory that makes sense on spec(Z[X]), the set of prime ideals of the ring of polynomials with integer coefficients, and under the Zariski topology!

There is however also, at least for non singular varieties over the complex numbers, which do have nicely varying tangent and cotangent spaces, an algebraic version of the deRham cohomology, i.e. the one defined using differential forms:
http://www.numdam.org/article/PMIHES_1966__29__95_0.pdf
"Algebraic geometry" is a big term, and many complex algebraic geometers work on what are essentially complex manifolds, where classical cohomology, and in particular Hodge's theory of harmonic forms, is a basic tool. Kodaira's embedding theorem in algebraic geometry for instance identified which compact complex manifolds can be embedded in projective space as the zero locus of polynomials. I worked all my career mainly on Riemann surfaces, which are not only complex manifolds, but one dimensional ones. Of course that led to the related studies of their Jacobian varieties, whose "theta divisors" are somewhat singular, as well as their "moduli spaces" which are quotients of manifolds by algebraic groups, and consequently also singular. All these related objects are also of higher dimension. Tangent vectors also play an important role in the study of singular varieties, where they give the "tangent cone", a fundamental tool for studying singular varieties, including classical ones.

The famous "Riemann singularity theorem" proved by Riemann himself, says that line bundles of degree g-1 on a Riemann surface S of genus g, and having at least one section, are parametrized by a subvariety Theta(S) of dimension g-1 in the Jacobian variety (a g dimensional complex manifold), and that a line bundle L has 2 or more independent sections iff the point of Theta(S) corresponding to L is singular. In fact the maximal number of linearly independent sections equals the degree of the tangent cone, which cone has degree one exactly when the point is non singular, in which case the cone is thus a linear tangent space, hence of degree one.

The best proof of this is due to Mumford and Kempf and is exposited in the appendix of this paper of mine and Robert Varley, where we attempt to at least partially generalize the result in some cases to some relative versions of the Jacobian, i.e. to the Prym variety of a double cover of Riemann surfaces.
http://alpha.math.uga.edu/%7Eroy/onparam.pdf

 

[mathwonk]

If we restrict ourselves to the classical setup, i.e. a classical algebraic variety over the complex numbers, or even over any field containing the rational numbers, there is a famous theorem due to Hironaka that allows us, for some purposes, to reduce its study to that of a non singular variety, i.e. a manifold. First of all it is known that as long as our variety is not considered with positive "multiplicity" globally, i.e. is "reduced and irreducible", (this means for a variety defined by one equation, that the equation is irreducible, in particular not a power of another equation), then there is an open dense subset of the variety whose points do look like a manifold, i.e. the "non singular, or smooth, locus" is open and dense. So every variety is a manifold almost everywhere, since the singularities take up very little space, e.g. they have measure zero. But Hironaka's theorem goes much farther and allows us to smooth out the singular points as well. I.e.

Theorem: If Y is any reduced and irreducible algebraic variety over say the complex (or real) numbers, there is a non singular variety X and a morphism f:X-->Y which is an isomorphism over precisely the non singular points of Y, i.e. f^(-1)(Ysmooth) = Xsmooth. Moreover f is a proper morphism, in particular f is surjective, and the inverse image of the singular locus of Y is a union of submanifolds of X of codimension one; moreover their intersections are as simple as possible, looking locally like some of the coordinate hyperplanes of Euclidean space meeting at the origin.

This is called "strong resolution" and here is a lecture for algebraic geometers explaining it, by Ja'nos Kolla'r. The introductory parts of this lecture are surprizingly readable, as is Kolla'rs style, but it helps to know that "birational" means "isomorphic over a dense open set", or equivalently, "having the same function field". Also an ideal is principal if it has only one generator, so principalizing the ideal of a subvariety changes it into a subvariety defined by one equation, hence into a subvariety of codimension one, also known as "blowing up". E.g. the map (x,y)--->(x,xy) = (s,t), pulls back the origin (s,t) = (0,0) in the (s,t) plane, a set of codimension two, to the set x = 0, a set of codimension one in the (x,y) plane. Geometrically this map is an isomorphism from the complement of the line x=0, to the complement of the line s=0, but the whole line x=0 gets squashed to the point s = t = 0. This is a "blowing down" and viewed in the backwards direction, is considered a "blowing up". I.e. this map blows up the origin of the (s,t) plane, to the line x=0 in the (x.y) plane.

If we restrict attention to what this map does to the curve t^2 = s^2(s+1), which crosses itself at the origin, (see the picture in the middle of the page at the right)

https://en.wikipedia.org/wiki/Cubic_plane_curve
we get, putting x=s, yx = t, the pullback curve x^2y^2 = x^2(x+1), or x^2(y^2-(x+1)) = 0. This last curve has two parts, y^2 - (x+1) = 0 , which is isomorphic to the original curve except for having now two points, (0,1) and (0,-1), that map to the singular point of the original curve at the origin, and the doubled line x=0, that collapses to the origin. Throwing away this line we are left with the desingularized version y^2 = x+1, of our original curve, t^2 = s^2(s+1), obtained by blowing up the origin. Hironaka showed in 1964 this could be done for any variety, in an Annals paper maybe 200 pages long, and won the Fields medal. Kolla'r discusses some of the simplifications in the proof obtained over the next 50 or so years.

https://arxiv.org/pdf/math/0508332.pdf

 

[member 587159]

@mathwonk I prefer a book that covers every detail, even if it means more than 1000 pages to read than a slim book that doesn't.
Especially if those "gaps" in those books are hard to complete by yourself!

Agree. Exercises should be in the exercises section, not in the main body of the text,unless it is a routine verification or something like that.

 

[mathwonk]

I appreciate this sentiment. I want to suggest however that we do not always have a choice of using only books with all our ideal qualities. E.g. one wants not only details enough to be clear to us, but also that the results should be correct, and the choice of topics should be authoritative. I.e. even if the details are hard to process fully in a given book, it can still be useful to read just the table of contents to find out what is deemed important, assuming of course the author is a real authority. Unfortunately for many of us these authoritative authors tend to take for granted many facts we may not know. But on the other hand, books filled with details tend to be written by people like me, who are trying to fill details for themselves in books they found hard to read. Still we are dependent on those more difficult authors to know what topics to include.

So I suggest reading foremost the books written by real experts, and supplementing them by reading the details in other books. But if one reads only the more detailed books by secondary authors, one runs the risk of missing out on the deep insights available only from the most authoritative authors.

I don't want to pick on anyone in particular, but I have recently perused some very detailed books on algebraic geometry that are very highly praised here and elsewhere for their readability, but that seem to contain significant errors due to lack of deep comprehension of some basic concepts. E.g. some such books claim that a regular map from a projective variety V to projective n space, is given by a sequence of n+1 homogeneous polynomials of the same degree, with no common zeroes on V. This is false in general, and such maps define only maps with finite fibers. Some such books even seem to assert that such maps are defined by a sequence of n+1 regular functions with no common zeroes on V. Then on the next page they point out that all such functions are constant!

Another remark is that Mumford makes a point in his red book to emphasize the key result called Zariski's main theorem, whereas there is not even a mention of it in the more detailed book by the less expert author.

Mumford may leave much to be mulled over, but he does understand the topic, and whatever he says you should know, probably you should try to grasp it. I learned a lot from Riemann, a notoriously hard author to understand, just by reading the headings of the paragraphs in his great paper on abelian functions. So I agree it is hard to learn a subject well from only a book that leaves many gaps, but it is perhaps impossible to learn it deeply from a source whose author does not actually understand it deeply.

I agree also that the book EGA is both detailed and authored by experts, so maybe some of you will benefit from it more than I have myself. I am still motivated to recommend using Mumford as a guide to EGA. And thank you! Maybe now I will take my own advice and try it again myself! (But of course recall again that EGA was never completed, so using it as ones only source, even for those authors' most significant results, is not an option.)

Another book I have seen only in very brief excerpts online, but that is very well liked by amazon reviewers, is the following one, which seems to have great detail, and is written by professional arithmetic geometers: It is also on sale at 40% off from springer for the next couple weeks.
https://www.springer.com/us/book/9783834806765Since I recommend books by experts, I want to mention one nice looking source that is much more elementary than EGA, indeed does not treat schemes at all, but is a first introduction to algebraic geometry by a top expert who seems to enjoy the challenge of writing for beginners, Mike Artin of MIT, (you may be familiar with his outstanding undergraduate abstract algebra book): warning this website may not be secure.

http://math.mit.edu/classes/18.721/agnotes-jan18v13.pdf

Here is another similar recommendation, an introductory algebraic geometry course by a high level expert, James Milne, but written for beginners. The website also seems secure:
https://www.jmilne.org/math/CourseNotes/AG.pdf
Both these courses are written by people who understand schemes deeply, and are writing elementary course notes in a way that is designed to make the later transition to schemes a natural one.

 

[mathwonk]

I have recently had occasion to read some of the notes from Ravi Vakil at Stanford, linked in post #37. I found them excellent indeed, and now think they may be the best source for someone wanting to learn the Grothendieck style methods and ideas. Ravi's writing is completely unlike much writing from the 1960's, in that it is oriented toward the learner, and full of tips for how to understand the material. Of course I also find Mumford's writing to be full of tips for understanding, but I admit his redbook required rather more algebraic background than I had available, and in the years I have been reading it I have had to take breaks and read whole books on commutative algebra, such as a chapter on field theory from Zariski - Samuel, and the whole book Undergraduate Commutative algebra by Miles Reid, and also the chapter on commutative algebra from Dummitt and Foote. Of course I have enjoyed and benefited from this reading.

For those interested, Ravi posted a number of "pseudolectures" on his notes, on youtube:

 

[FourEyedRaven]

Hi.

This is a short guide to learn the essentials of algebraic geometry up to the language of sheaves. Here are lecture notes for an introductory course, and for a second course, in algebraic geometry. Both these notes are highly regarded.

Undergraduate Algebraic Geometry - Reid (134pg)
https://homepages.warwick.ac.uk/staff/Miles.Reid/MA4A5/UAG.pdf

Algebraic Geometry - Gathmann (138pg)
https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2019/alggeom-2019.pdf

Algebraic Geometry is heavily based on abstract algebra and Gathmann's notes, especially, make heavy use of it, particularly commutative algebra. For those who do not have a background in abstract algebra, here is a crash course with material up to commutative algebra.

Groups - Dexter Chua (56pg)
https://dec41.user.srcf.net/notes/IA_M/groups.pdf

Groups, Rings, and Modules - Dexter Chua (96pg)
https://dec41.user.srcf.net/notes/IB_L/groups_rings_and_modules.pdf

Commutative Algebra - Gathmann (132pg)
https://www.mathematik.uni-kl.de/~gathmann/class/commalg-2013/commalg-2013.pdf

Cheers

 

[mathwonk]

I would also recommend Miles Reid's undergraduate commutative algebra.

 

[FourEyedRaven]

Hi.

This is a short guide to learn the essentials of algebraic geometry up to the language of sheaves. Here are lecture notes for an introductory course, and for a second course, in algebraic geometry. Both these notes are highly regarded.

Undergraduate Algebraic Geometry - Reid (134pg)
https://homepages.warwick.ac.uk/staff/Miles.Reid/MA4A5/UAG.pdf

Algebraic Geometry - Gathmann (138pg)
https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2019/alggeom-2019.pdf

Algebraic Geometry is heavily based on abstract algebra and Gathmann's notes, especially, make heavy use of it, particularly commutative algebra. For those who do not have a background in abstract algebra, here is a crash course with material up to commutative algebra.

Groups - Dexter Chua (56pg)
https://dec41.user.srcf.net/notes/IA_M/groups.pdf

Groups, Rings, and Modules - Dexter Chua (96pg)
https://dec41.user.srcf.net/notes/IB_L/groups_rings_and_modules.pdf

Commutative Algebra - Gathmann (132pg)
https://www.mathematik.uni-kl.de/~gathmann/class/commalg-2013/commalg-2013.pdf

Cheers

I would also recommend Miles Reid's undergraduate commutative algebra.

Good suggestion.

When I said "sheaves" above, I should have said "schemes". Also, I forgot to add that general topology plays an important role too, but an in depth knowledge of general topology is not necessary. So I link these lecture notes that introduce

Metric Spaces and Topology - Dexter Chua (41pg)
https://dec41.user.srcf.net/notes/IB_E/metric_and_topological_spaces.pdf