12:07 AM
I realized I don't actually know how to orient $\mathbb{C}\mathbb{P}^2$. How do I do that?

All complex manifolds have a canonical orientation.

It's a complex manifold, comes with a natural orientation.

If you want bases, use $v,Jv$. If you want a differential form, take $(\sqrt{-1})^n dz_1\wedge d\bar z_1\wedge\dots\wedge dz_n\wedge d\bar z_n$.

@Thorgott @feynhat Geometric Topology Auto IV: Donaldson Theory

Is there any reason as to why there was less mathematical rigor during the Renaissance?

12:17 AM
There weren't the tools and ideas for the rigor we can apply today

Rigor only makes sense in context of the language math is spoken. We consider the language of mathematics during Renaissance primitive, and we will be seen as primitive in the future.

Cauchy was the one who formalized limits. That was rather late.

urgh, how do I complex differential forms

So then is something like Euclid's Elements considered rigorous, even though it was made millenia ago?

Yes, quite.

12:18 AM
@Balarka lol, delete the auto and that could be an actual book title

It was surprisingly ahead of time.

Makes one wonder how "rigor" will be a thousand, or even a hundred years from now.

If we survive such a long stretch of time and people still do mathematics that is

After the coronavirus destroys the earth, who knows.

O_O True.

12:21 AM
Elements is a weird case because that's a text that's been kind of used as the gold standard for all rigor to come after it (in the West).
Axioms, postulates, definitions -> proposition -> proof -> proposition depending on previous -> proof -> etc

The foundations in Elements was criticized almost 2000 years after Euclid, by Gauss, when he discovered hyperbolic geometry. During that time foundations in geometry was nearly unchanged. Immediately afterwards Klein developed a new context for geometry which is used still to date.
It's taken a long time to break through the standards set by Euclid.
And as Fargle says, we still use that kind style of writing in mathematics.

The fact that it took about 2000 years to confirm Euclid's intuition that you needed the parallel postulate to do ordinary flat geometry speaks volumes for its longevity

Hard to beat Euclid and Gauss.

So then was Elements more of the type of text that was ahead of its time, or was it just that everything after that and before the 18th century was a regression of sorts?

Same difference, right?
If an era is lagging we don't call time was behind
If a person did something that held up through that era we say the person was ahead of time

12:29 AM
Of its time more than ahead---Greek mathematics, as far as I recollect, represented the latter half of a slow and subtle shift in Western mathematical traditions from concrete to abstract. Don't quote me on that though, I'm not a math historian

I don't know dude, that axiomatic setup is more abstract to me than what the Indians and later Italians were doing with theory of equations

that trend being in antiquity, I mean
Like, comparing later Greek mathematics with earlier Greek/Babylonian/Egyptian mathematics (although our sources on Egyptian math are scant)

Sure yeah I get your point

But yeah, I wouldn't call Renaissance math a regression, just less rigorous

Nah there were definitely major progressions made (hail elementary algebra).

12:35 AM
Depending on your definition of Renaissance, calculus too.

Infinitesimals and whatnot, yes.

Yeah, Italians made massive progress on theory of equations and what we call classical algebraic geometry now
Calculus was indeed developing bit by bit
Omar Khayyam questioned the parallel postulate actually. Also he did stuff on cubic independently of Cardano

ye

Didn't he solve it in terms of the parabola and a circle or something?
I don't recall the specifics.

It was some geometric method, yeah. I don't recall specific either. It's probably true that contemporary mathematics in Islamic tradition already made advances when Europe was too busy with the crusades

12:41 AM
lol

Yeah there was definitely less mathematical output in Europe during the medieval period, at least that has endured

The scientific revolution in 17th century was the big comeback
Starting with Galileo
It's important to note that there are adverse effects of rigor too. Grassmann developed an abstract theory of exterior algebras that a man of high stature such as Kummer didn't understand
In the early 19th century

the adverse effects of rigor are 20th century mathematics
smug face

The exposition was too nonrigorous, he said, IIRC
Of course the Galois example also fits the bill but that's been reiterated too often
Ah Grassmann apparently wrote a second textbook on linear algebra which was completely rigorous but it was still ignored by and large by the scientific community

Where does the 21st century stand on this subjective rigor scale, then?
I mean there is so much going on math-wise that it'd probs be too hard to summarize.

12:50 AM
if I had to make a guess I'd say we're still majorly using the formal groundworks set up in 20th century
but its way too early in 21st century to say anything, right?
Gromov suspects a new post-Kolomogorov language for probability will be developed during this century
Which seems like a good guess to me
Homotopy theory has seen major changes in language in the last few decades

Sound advanced solely by the names, lol.

I mean it's been 20 years since 20th century. Very early to see major changes except in specialized areas
Usually a century or two later people will be able to give an accurate summary of the important changes made

@BalarkaSen I'd certainly hope so. Alas, it's hard for the regular human being to parse gromov's musings about ergosystems and such

And it'll still be too specialized. At least half a millennium later average undergraduates in mathematics will hear stories about how we were idiots
@user2103480 Yeah hahah
I tried to read "In Search Of A New Structure", which is about entropy, but gave up 10 pages later

Also, as an outsider, it's hard to see whether the works of people inspired by gromov's approach to entropy are actually useful

12:58 AM
Also I completely ignored his category theory formalism because it obscures the point more than clarifying it to me

@BalarkaSen Oh boy when he starts with the ergodic theory/symbolic dynamics I get thrown out the window

I liked his exposition of the proof of isoperimetric inequality without sharp constants using entropy
It did make sense that the Kolmogorov point of view was obscuring the main idea
But things like finite probability spaces, discretizing general probability measure spaces (what he calls the subcategory of finite prob spaces being dense in the topological category of probability measure spaces) are well-known to computer scientists

@BalarkaSen do you mean borel codes and other things for computable measure theory?

I was thinking of VC dimension
I have seen successful proofs of things in geometric probability by discretizing and using probabilistic methods on finite spaces
To clarify, I am not an expert. I was taught by colleagues who are

Ahh. Ugh. I'm damaged by CS classes, just seen it in the context of PAC learning and I really didn't know what to make of it

1:09 AM
Haha I see. You're a CS student?

Nah, math, but I took ML classes to have a "backdrop" lol

Gotcha
What kind of math are you into

Mixed. Started specializing in logic, up to forcing and such, now shifted more towards probability theory & analysis, but taking a course on semantics of HoTT on the side which is giving me headaches

Yikes, bizarre mix
We have a few logic and set theory people here. @Alessandro, in particular, who knows HoTT

Haha, I actually took a class with him

1:14 AM
Oh

small world darn

On set theory of course
Ah and one about HoTT, but that was just about reasoning inside the system

Scary.

@BalarkaSen But yeah, it's not working in my favor that these 3 topics are almost disjoint in many relevant topics

Understandably
Oh didn't you ask for access in garbology a while back? I forgot about it; fixed now.

1:18 AM
Yeah to bother you guys with questions on simplicial sets

Lol I see
I forgot whatever I learnt once.

Understandable, it seems like the kind of topic. Trying different literature as side references is a mess since the expositions differ a lot in structure, notation, and approaches to proofs

Yup. Are you using Goerss-Jardine or something?

I looked at that and was horrified. Mostly, I use Rezk's intro to quasicategories, which I find pretty understandable, but doesn't cover everything in the course

Gotcha.

1:23 AM
to be precise: it covers a lot more, but not the same topics

I just never ended up using any substantial simplicial homotopy theory. I think all I wanted to understand was if $G$ is a simplicial group acting freely on a simplicial set $X$ then $X \to X/G$ is a Kan fibration or some variant of this. This simplifies a lot of fact-checking when dealing with fibrations of space of immersions, embeddings, diffeomorphism groups of manifolds, ...

Having skipped straight from no algebraic topology to simplicial sets, which is more manageable than expected, books such as Goerss-Jardine expect too many prerequisites
@BalarkaSen You're working in GGT I gathered?
broadly speaking

I don't work on much, I am an undergrad. I like thinking about things, but mostly I think about topology.
I have been recently thinking about probability that's above my paygrade

@BalarkaSen Hahaha what do you mean

first passage percolation to be specific

1:28 AM
darn that all seems so advanced from the perspective of a freshman undergrad
It almost feels like there's a point where you know math on a deep level, somewhere halfway thru the major.

Percolation is heavy. In my introductory course on martingales and ergodic theory, we were given a few exercises that scratched the surface of percolation theory, since that is what the professor and assistants specialized in

@Naganite There's no point, you either do math to have fun (like playing video games) or get a career (like corporate jobs).
The latter is more monetizable than the former

Obviously, in the end it's all about having fun while doing something that allows you to eat and sleep every day.

Those were hard both technically and conceptually

@user2103480 Ah I know very little about either martingales or ergodic theory

1:32 AM
@BalarkaSen I was referring more on a conceptual level; I know Cal 1 and I can understand how to understand some stuff beyond that like Linear Algebra, Multivar Cal, and ODE, but then beyond that it just looks like alien language.

I assume you were asked to prove the velocity of percolation exists a.e.? That's a corollary of subadditive ergodic theorem
@Naganite Ah OK. Yeah, as you learn more you start guessing what words mean better

@BalarkaSen it's interesting once you're used to the theory. Most applications are in finance or physics though, which isn't so lucky for me. But I'm taking stochastic PDE soon from a prof that specializes in stochastic neuroscience, so that might be a cool topic to dive in
@BalarkaSen but it will always stay an alien language once you move away from your comfort zone

Nice! I don't know anything about SPDEs
True, @user2103480

@BalarkaSen Let me check, I don't think that was it. Maybe I overstated

Any of you all know some Galois Theory?

1:36 AM
Man I don't know anything about these damn inequalities in probability. So confusing.
I can't tell the conceptual difference between $<$ and $>$ in half of these inequalities

@Naganite I learned enough to know that I want to stay away from it

Sweet jesus what is it even saying???
I was asking, primarily because I was looking into univariate polynomials for months and months, and it took me a while, but I eventually figured out how to invert a cubic ($y = ax^3 + bx^2 + cx + d$) with some substitutions.

Quadratic is just a linear substitution, cubic was a weirder rational one, and IDK quartic. How come you can't do some substitution magic with a quintic though?

Yeah (b) is existence of velocity in FPP

Ahaha ok there you know more than me

I tried checking Math.SE for answers but all I could really understand is that it has to do with symmetric groups.

1:42 AM
@Naganite Yeah it's pretty confusing why quintics don't work. Somehow the point is 5 things have more complicated symmetries than 4 or less things.

wack

It's cool that you managed to solve a cubic! I honestly don't have a clue except something something permutation groups

Even then the intuition behind it is beyond me.

@BalarkaSen Btw when you made that meme about throwing finite groups into the bin

I felt that

Hahah

1:44 AM
I will say that in the end it is somewhat arbitrary, because (IIRC) Abel-Ruffini Theorem says the general quintic is unsolvable in terms of elementary operations and $\sqrt[n]$ roots.

@user2103480 It seems this is some concrete proof avoiding Kingman's theorem.

Did that format incorrectly? :(

Try \sqrt

mission failed

@Naganite At first, it says a lot of things that are easily explained by a picture but ugly in text

1:44 AM
$\sqrt{n}$

I was trying to put nth root.

But yeah.

$\sqrt[n]{m}$
$\sqrt[n]{}$
\sqrt[n]{}

Apparently using some substitutions, you can boil the general quintic down to $f(x) = x^5 + x + a$, but that's it.

1:45 AM
yeah oh man

Also, how did you type it out without it being formatted?

The exercise statements are harder to explain though

So then the solution to that is just define $f^{-1}(x)$

...as an operation, LOL.

bring radicals

@BalarkaSen We cited kingman's theorem without proof and I think it was applied here and in the preceding exercise

1:46 AM
BR moment

lolol
@user2103480 Ah ok
Yeah it would be surprising otherwise. Kingman discovered his theorem in the process of establishing this exercise
:P

Apparently quintics can be solved with things called elliptic integrals and trig functions.
I tried taking a peak and instantly ran away.

Try reading R. B. King's book on quintics

@BalarkaSen Sounds like a harsh problem set!

That was my first exposure to advanced mathematics
@user2103480 Yeah haha

1:49 AM
I'll head to sleep now, cheers!

See ya!

ðŸ‘‹

I'm gonna bail out and try to see if I can make head or tails of probability
The coin is unbiased
:3

And then just when you think that Abstract Algebra was tough as is, you look at bivariate polynomials. O_O
AFAIK that has to do with algebraic geometry but the name is the limit of my knowledge.

2:22 AM
What does "order matters" mean in 5. and 6. of the twentyfold way (en.wikipedia.org/wiki/Twelvefold_way#The_twentyfold_way)? My understanding is that the twelvefold way (which doesn't include these two) already covers ordering VS nonordering of both the domain and the codomain. What additional order is being considered?

3:04 AM
@Yuvraj I don't see that exact question on MSE. I have written up an answer and an illustration in case you write it up as a question.

hey fam
assuming i've done calc 1 & 2 only
can i learn abt the lagrange inversion theorem and the lambert W func?
what more do i need to learn to be off with enough prerequisites?

1 hour later…
4:16 AM
@EdwardEvans Can you tell me how letters can take articles? An example from English will be fine.

"Das A steht für Albrecht Herzog von Bayern" : "The A stands for duke Albrecht of Bavaria"

@LeakyNun Okay! All it means is that we do have three articles in German, but the letters take only das, right?

(well there are much more)

How many languages do you know?

the articles change form depending on their role in the sentence
subject / object / indirect object / possessive

4:23 AM
Yes

the Latinate terms are Nominativ / Akkusativ / Dativ / Genitiv

Okay
So, there we go you know: English, German, Latin and probably Chinese, right?

2 hours later…
6:09 AM
Let \begin{align*} f(x,y) := & \quad \sqrt{x^2 + y^2} \qquad & \text{if $x$ or $y$ is irrational.} \\
& \quad \quad \quad 0 \qquad & \text{otherwise.} \end{align*}
Can anyone help me out and verify if this is a counterexample to the question I asked here:
1

Suppose we have a function $f : \mathbb{R}^m \rightarrow \mathbb{R}^n$ with the condition that the rows of the Jacobian matrix of $f$ are linearly independent on $B \subseteq \mathbb{R}^m$. Let $B_p$ = $\{ t \in B : \forall s \in \mathbb{R}^m \; \; \lvert s - t \rvert < \frac{1}{p} \implies \fra... Setting$B = \{(x,y) : x \text{ and } y \text{ irrational} \}$1 hour later… 7:41 AM hey can anyone answer me? :| $$\dfrac{x^2}{(x-1)^2}+\dfrac{y^2}{(y-1)^2}+\dfrac{z^2}{(z-1)^2}=\left(\dfrac{x}{x-1} +\dfrac{y}{y-1}+\dfrac{z}{z-1}-1\right)^2+1$$ how to think,to derive at this?! intuition? @user69608 Don't you need some additional conditions for that to be true? Maybe$xyz=1$? How is it humanly possible to derive this? How to derive the fact that $$\dfrac{x^2}{(x-1)^2}+\dfrac{y^2}{(y-1)^2}+\dfrac{z^2}{(z-1)^2}=\left(\frac{x}{x-1} +\dfrac{y}{y-1}+\dfrac{z}{z-1}-1\right)^2+1$$I can prove this by expanding, but there's no way on earth I would've found that equality myself. — user26486 Dec 23 '13 at 21:38 @MartinSleziak oh yes sorry forgot that okay :( i'll just say it again i've done calc 1 & 2 only; can i learn abt the lagrange inversion theorem and the lambert W func? what more do i need to learn to be off with enough prerequisites? 8:20 AM "Let$\mathfrak{m} \in \text{Max}(R)$and$\mathfrak{p} \in \text{Spec} R$. Show that$(R, \mathfrak{m})$is a local ring iff$1 + \mathfrak{m} \subset R^{\times}$." this pset has lots of unnecessary junk in the problem statements lmao weird af first exercise had the weird condition that$S$be an$R$-algebra and now this @Alessandro Do you know much about the geometry of Heisenberg group? A presentation seems to be$\langle a, b | [[a, b], a] = 1, [[a, b], b] = 1\rangle$Here's a picture of the Cayley graph.$z = [x, y]$, I believe. Doing a commutator along$x$and$y$climbs you up the$z$-axis Ah the geodesics literally swerve upward like Lagrangians in a contact manifold. This is what they mean by Carnot-Caratheodory metric And this has to do with commutator length I am sure. Nutcases, Gromov and his student Pansu Flag, seriously? 8:43 AM wasn't me, I'm just drinking my coffee. The flag brought me here. Just wanted to say that I declined it. "Let$\mathcal{A}_{\text{Zar}}$be the set of closed sets of a topology$\mathcal{T}_{\text{Zar}}$... " I also don't get why there's a distinction being made between A_Zar and T_Zar who wrote this pset @Knight We have a stack for that: german.stackexchange.com/q/17689/9551 I mean that seems like a fair notation @Edward But I'm used to the topology itself being the set of open sets/closed sets 8:51 AM Yeah T is usually reserved for the collection of open sets which define the topology Isn't the Zariski topology usually defined by its closed sets tho? It's kind of a stupid point but whatever Yeah, hence why they bother with A at all Indeed lol Algebra is backwards so closed sets define the topology not the open sets right 8:53 AM This is a general theme of algebra. Everything is backwards I mean it's not confusing, but there's so many problems on this pset that have unnecessary information in the statements rofl Even when you make some mistake it's because of a forward/backward issue. Maybe pushforward became pullback, or a left arrow because a right arrow in your diagram, ... either that or I've done literally everything wrong and I'ma get 0 on this pset hahaha That$\mathfrak{p}$thing seems to verify you're correct lol 8:55 AM I mean literally. You want to say a variety is contained in some other variety, but the inclusion of ideals go the other way You want to get a map of varieties$X \to Y$but suddenly you realize you've been writing maps$k[X] \to k[Y]$and everything you did was wrong$I \subset J \implies V(J) \subset V(I)$dis?$\text{Geometry}^{op} = \text{Algebra}$is the meta theorem Yes, @Ed hahaha meta woa You had nice covariant functors. Boom son they're all contravariant now nah they're still covariant, you just opped the cat 8:59 AM I'm gonna bail out, got a probability meeting with someone. Need to figure out some last minute details Cya The problem with analysis is$\text{Analysis}^{op} = \text{Analysis}$, inequalities don't make sense.$<$is the same as$>$9:10 AM @BalarkaSen I'm afraid I don't even know what the Heisenberg group is it's a conglomerate of meth vendors 1 hour later… 10:23 AM Sets of the form$U(f) := \lbrace \mathfrak{p} \in \operatorname{Spec} R : f \notin \mathfrak{p}\rbrace$form a basis for the Zariski topology; if$\mathfrak{p} \in U(f) \cap U(f')$then$\mathfrak{p} \in U(ff') \subset U(f) \cap U(f')$and i think$\bigcup_{f \in R\setminus I} U(f) = V(I)$@Thorgott Thank You If 2 gaussian distributions have same mean and same standard deviation; would their graphs be identical (superposable)? Or still they can "cut" each other in some ways? 11:23 AM two Gaussian distributions with the same mean and same standard deviation are in fact the same distribution a Gaussian is uniquely determined by two parameters, its mean and its standard deviation 11:40 AM @EdwardEvans consider idk$I = 2 \Bbb Z$or any example at all 11:59 AM @FadedGiant Thanks Loong, I shall surely join it. 12:35 PM @Leaky yeah i was being a retard, got it now lol Tyty 12:49 PM @Naganite I don't have a full answer to this, but maybe a useful comment: If you have a quadratic polynomial$X^2+pX+q$, you can substitute$X\mapsto X-\frac{p}{2}$to bring it to the form$X^2+(q-\frac{p^2}{4})$. This substitution is precisely done so that the$X$terms cancel out. Similarly, if you have a cubic$X^3+aX^2+bX+c$, you can substitute$X\mapsto X-\frac{a}{3}$to get the cubic$X^3+(b-\frac{a^2}{3})X+(\frac{2a^3}{27}-\frac{ab}{3}+c)$. Indeed, there is a pattern here. If you have a monic polynomial of degree$n$, substitute$X\mapsto X-\frac{a}{n}$, where$a$is the coefficient 1:40 PM @Thorgott Wowwww that is so interesting @Thorgott Thank you so much 1 hour later… 3:05 PM @Thorgott just thought, my argument for the injectivity of$S^{-1}N$given an injective$N$was wrong yesterday, since$\operatorname{Hom}_{S^{-1}R}(S^{-1}(-), S^{-1}N)$is contravariant, so you actually get a surjection $$\operatorname{Hom}_{S^{-1}R}(S^{-1}R, S^{-1}N) \to \operatorname{Hom}_{S^{-1}R}(S^{-1}I, S^{-1}N)$$ but that's actually what I want so thatÄs fine just that my conclusion was incorrect lol oh, of course, you're right this is contravariant hom indeeeed so this says that every morphism$S^{-1}I \to S^{-1}N$comes from a morphism$S^{-1}R \to S^{-1}N$, which is what I want for Baer's criterion to work rather than my conclusion yesterday, which was that a morphism$S^{-1}I \to S^{-1}N$literally is a morphism$S^{-1}R \to S^{-1}N$lol 3:38 PM why is this surjection the restriction map, actually? probably just need to chase definitions 2 hours later… 5:20 PM Hi to everybody and nobody A while ago I was reading Frobenius theorem (manifold) and although I didn't understand it completely, a key takeaway was the existence of integral manifolds which was used to prove the existence of solution on 1st order homogenous linear PDE. This got me thinking that to prove the higher order inhomogenous case, we (if possible) can extend this theorem for a tensor field. Is there any research/result in this direction. 5:36 PM could someone enlighten me on what "unitary" means in this context? I think it means that the space is equipped with a complex inner product Ah I see, thank you 1 hour later… 6:43 PM I have to find the sum of the roots of$tan^2(x) - 9tan(x) + 1 = 0$between$0 <= x < 2\pi$. I've ended up with$\frac{n([(n-1)\pi + arcsin(\frac92)] + [(n-1)\pi + 2arctan(\sqrt{\frac{81}{4} - 1} + \frac{9}{2}])}{2}$but apparently It's doable without a calculator as well. Could I please get any hints? @VioAriton If you do a substitution you get a polynomial (so the term roots actually makes sense). Then use that the sum of the roots can be read directly from the coefficients. @henceproved The generalizations of Frobenius were started by Cartan and Kähler a century ago. Look up the Cartan-Kähler theorem and exterior differential systems. I recommend the relatively recent book by Bryant, Chern, Gardner, Goldschmidt, Griffiths with the title Exterior Differential Systems. But it's quite advanced. 7:17 PM Do mathematicians really understand what a torus is? Do mathematicians really understand anything? I think the answer to these questions should be the same, no matter which one it actually is 7:33 PM at least mathematicians know that$\Bbb R^2$is homeomorphic to$[0,1]^2$Stop saying nonsense. not this again I thought you said$\Bbb R^2$is something to$(0,1)^2$could have been diffeomorphic who knows yeah,$\Bbb R^2$is diffeomorphic to$(0,1)^2$7:50 PM If [ and ( are homeomorphic, how can (0,1) and [0,1] not be? 2 can't I make mistakes? surely, there are examples of spaces such that$X^2\cong Y^2$, yet$X\not\cong Y$@Thorgott I'm 150% confident the answer is yes, but I don't have an example right now 8 For topological spaces$X$and$Y$, is it possible that$X \times X$and$Y \times Y$are homeomorphic, but$X$and$Y$are not homeomorphic? (I poked around with finite spaces, and manifolds, and the Cantor set, without seeing any examples.) This was inspired by Existence of topological space ... huh, I'm surprised this is non-trivial 8:07 PM There's also funny things like spaces$X\cong X^n$but not for smaller powers That reminds me of a fun one I cooked up an answer to 29 Does there exist a topological space$X$such that$X \ncong Y\times Y$for every topological space$Y$but $$X\times X \times X \cong Z\times Z$$ for some topological space$Z$? Here$\cong$means homeomorphic. That's some very weird stuff it's weird that in both of these cases, the counter-examples are manifolds and some machinery is required to see they satisfy the requirements intuitively I would expect that it's easier to find counter-examples in a less well-behaved subcategory But on the other hand there's much more machinery available for good spaces True, I guess restricting yourself to a subcategory you can actually say something about gives a way of somewhat systematically ruling out which spaces will and which spaces won't work/what properties they must satisfy. When working in the general topological category, you don't really have many options other than guessing. especially in the second case, I think "being a square" is probably a very hard property to detect in general 8:20 PM If you start looking at things like totally disconnected compacta I think you have a lot more flexibility The Cantor set is its own nth root, say @MikeMiller that's a thing zero dimensional spaces tend to do Yeah, I know, I just mean that like when you get into the next simplest kind of not-that-simple spaces they are harder to control There's also positive dimensional spaces homeomorphic to their square though, they are used to show that the inequality$\dim(X\times Y)\leq\dim X+\dim Y$can be strict in the product theorem for the covering dimension Awful The subspace of$\ell^2$made of points with only rational coordinates (this is sometimes called the Erdos space) has dimension 1 and is homeomorphic to its square 8:27 PM That's awful And you can actually look at the subspace made of sequences all of whose coordinates are of the form 1 over an integer, the closure of that is still 1 dimensional, but now we even have a completely metrizable example Those are totally disconnected spaces with positive dimension which is another weird combination of properties 9:16 PM hides in a sand dune I thought the desertification of California would take longer. Slightly. Hi, @Fargle. Howdy Hello, please explain me why$\inf\{\varepsilon>0, A\subsest B(0,\varepsilon)\}=0$implies that$\forall \varepsilon>0, A\subset B(0,\varepsilon)$What do you not see how to do? 9:28 PM how to deduce that$\forall \varepsilon>0, A\subset B(0,\varepsilon)$Well, you just restated the question. I'm asking you where you get stuck? I don't know how to do how to start What does the inf statement tell you? $$inf X=0\Longleftrightarrow \forall \eta>0, \exists x\in X, 0\leq x<\eta$$ So pick an$\varepsilon>0$and proceed. What letter in your previous sentence should$\varepsilon$now become? 9:45 PM \eta ? Right. then$\exists x\in X, 0\leq x <\varepsilon$So write what that means.$X=\{\varepsilon >0, A\subset B(0,\varepsilon)\}$, then$x=\varepsilon$? Huh? That's not at all what your sentence says. 9:55 PM start ChatJax I don't see what satistisfy$x\in \{\varepsilon>0, A\subset B(0,\varepsilon)\}$You just know there is some$x$in the set that is$<\varepsilon$. You have no idea what it is. But write down what it means for$x$to be in that set. You should not be switching to$\varepsilon$s in the definition of the set. We have a fixed number$\varepsilon$. You don't get to put it everywhere. I don't know$A\subset B(0,x)$? Anyone knows where i can find estimating for the error when we estimate$\pi(x,q;a) \approx \frac{Li(x)}{\phi(x)}$Yes, you do know that. You're assuming everyone knows what all your symbols are, @user265855? 9:59 PM Where$\pi(x,q;a)$count the prime less than x that is congruent to a modulus q.$\phi(x)$is the ruler totient function, and$Li(x)$is the logarithmic integral This should be standard analytic number theory, but I do not know this field. The prime number theorem implies$\pi(x) = Li(x) + exp(xe^{a\sqrt{\log{x}}})$as$x \to \infty$, but i'm not sure how to use that to give a bound on$|\pi(x,q;a) - \frac{Li(X)}{\phi(q)}|$that's asking for quantitative bounds on Dirichlet's theorem, no? Well, from what i understand, Dirichlet's theorem is about the infinitude of primes in arithmetic progressions. I've been told to look at Brun-Titchmarsh theorem but I don't see how to use B-T to bound this error 10:32 PM Numbers that are congruent to a modulo q is precisely what an arithmetic progression is, so, in your notation, Dirichlet's theorem is the assertion that$\pi(x,q;a)\rightarrow\infty$as$x\rightarrow\infty$(when$a,q$coprime of course, nothing happens otherwise) and what you want is a quantitative result in that direction I don't know any analytic number theory, for the record, but that's the direction in which I'd look maybe you want something like this? en.wikipedia.org/wiki/Siegel%E2%80%93Walfisz_theorem 11:05 PM Thank you 11:16 PM I see Siegel-Walfisz theorem holds uniformly for any power of$\log{x}$which is quite restrictiive. *$q \leq (\log{x})^{A}$for some$A \geq 1\$