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1:47 AM
Hello, hello, hello, is there anybody out there?
 
I happen to be here.
 
@CLarue What do you happen do be doing?
 
hi folks! :D
 
Playing chess
a draw by repittion
 
2:10 AM
Hello all :)
 
hello
 
What's up?
 
@BenjaLim inverse images distribute through unions. I don't see any need to subtract an intersection.
 
Doin' some topology exercises here.
 
@Peter, what book?
 
2:13 AM
rep thry exercises
 
@AntonioVargas Introduction to Topology by Mendelson.
@anon How is it going?
 
@anon Are they fun?
 
just started on chap 2 exercises in fiona murnaghan's notes. some look like fun (the results are nice), others are just computation (do not want).
 
Out of all the computation I've had to do, the kind I did in algebra I enjoyed the least
I'll confess. I have nothing interesting to say.
 
 
@anon Failed to load! Darn.
 
(created after seeing this for the $n$th jillion time.)
 
@anon LAWL. My reading comprehension skills failed terribly here
 
@anon yes?
 
2:49 AM
@BenjaLim Ahoy.
 
maybe? what?
 
@anon Why should $h^{-1}(V) = g^{-1}(V) \cup f^{-1}(V)$?
 
@BenjaLim Isn't that trivial?
 
@PeterTamaroff ??????????
 
@BenjaLim I think it is.
 
2:51 AM
Wait, $V\subseteq X\cup Y$ right?
 
yes
 
I never said $g^{-1}(V)\cup f^{-1}(V)$ specifically I don't believe
 
Oh no
wait no
@anon No
$V$ is some subset of $T$
 
what's $T$?
 
where $f : X \to T$ and $g : Y \to T$
some other space
 
2:53 AM
oh, right.
 
because recall we want to prove there is a unique $h : X \cup Y \rightarrow T$
and I defined $$h(x) = \begin{cases} f(x), & x \in X \\ g(x), &x \in Y \end{cases}$$ @anon
 
Suppose $x\in X\cap Y$ such that $h(x)=t\in T$. If $x\in X$ then $x\in f^{-1}(t)$; if $x\in Y$ then $x\in g^{-1}(t)$. It must be the case that $x$ is in at least one of $X$ or $Y$ (or both).
 
@anon yes ?
 
$$\begin{array}{c l} x\in h^{-1}(T) & \iff h(x)\in T \\ & \iff (x\in X\wedge f(x)\in T)\vee(x\in Y\wedge g(x)\in T) \\ & \iff (x\in f^{-1}(T))\vee(x\in g^{-1}(T)) \\ & \iff x\in f^{-1}(T)\cup g^{-1}(T) \end{array}$$
 
that's so long
 
2:59 AM
@BenjaLim Depends on what you compare it to.
 
your math?
 
@BenjaLim You always remove stuff before I can see it!
 
@anon right so it is the case that $h^{-1}(V) = f^{-1}(V)\cup g^{-1}(V)$
and hence is open in $X \cup Y$
 
woops I guess I meant to use the letter V instead of T
 
nvm I got it
 
3:07 AM
Can there be sets neither open nor closed?
 
yes
 
Dread. OK.
 
eg (0,1] in Euclidean topology on R
 
Yes. OK.
 
Are there spaces in which all sets are either open or closed?
 
3:09 AM
I'm proving $\operatorname{int}(A)$, $\partial A$ and $\operatorname{int}(X\setminus A)$ are mutually disjoint.
 
obviously. the nontrivial version of the question is "what is the name or characterization of these spaces?"
 
@CLarue I think the discrete topology is an example of that.
 
@CLarue Put the discrete topology on any set.
 
wait, do you mean exlcusive or? if so, you're going to have to ignore the empty set and its complement
 
@PeterTamaroff the two interiors being disjoint is clear.
 
3:11 AM
@BenjaLim I already proved that
I'm trying to prove $\operatorname{int} (A)$ and $\partial A$ are disjoint now.
 
also, $\partial A= \partial (X\setminus A)$, so you only need to prove one of the other two disjointnesses
 
@anon Yes.
Well, I think I got it.
Reductio ad absurdum.
Suppose $x\in \partial A$ and $x\in \operatorname{int}A$
Then $x\in \overline A$ and $x\in \overline{X\setminus A}$
But $x\in \operatorname{int} A$ means $A$ is a nbhd of $x$.
Then $x\in O\subset A$ for some open set $O$.
But since $O\cap (X\setminus A)=\varnothing$
Drop the proof by contradiction.
Just use that $O\cap (X\setminus A)=\varnothing$
 
3:29 AM
It's funny that there is no compact notation à la "$\forall$" that means "for all but finitely many".
 
@MarkDominus We can invent it.
$\forall_{\not{\infty}}$
 
Yes, I was thinking of something like that.
 
maybe $\exists n\in \Bbb N: \exists ! \{x_1,\cdots, x_n\}\subseteq X: \neg P(x_i),i=1,\cdots,n$
 
That is a very generous notion of "compact" that you have there.
 
@anon Do you know what is the topology on $S^{\infty}$?
 
3:31 AM
@anon Never feed a dog with a tinfoil hat chocolate and put it in a microwave.
Also, never do that.
 
@BenjaLim is it metrical?
 
And you don't need the ! after the $\exists$ anyway.
 
@MarkDominus yes you do
 
@anon the topology on $S^n$ is coming from the euclidean topology
@anon but S infinity Idk
 
@MarkDominus $!$ means "unique" or something of the sort.
@BenjaLim Have you tried thinking about $S^{\infty}$ as a subspace of a Hilbert Space?
 
3:32 AM
I don't know what is a hilbert space
 
@BenjaLim It is not that hard of a definition:
 
If there is a finite sequence $x_1\ldots x_n$ for which $P(x_i)$ does not hold, then it does not matter if some of the $x_i$ are equal, since $P()$ still fails to hold for only a finite set.
 
a vector space that is complete wrt an inner product I believe
 
So you don't need the !.
 
Let $H$ be the set of all sequences $\{x_n\}$ of real numbers such that $\sum_{n=0}^\infty x_n^2$ converges.
 
3:33 AM
@MarkDominus if you don't have a $!$, then it doesn't bound the number of counterexamples to $P(\cdot)$...
 
@anon inner product gives a norm that gives a metric that is complete, such is a hilbert space?
 
yes
 
Define a metric on $H$ by $d(x,y)=\sqrt{\sum_{n=0}^\infty (x_n-y_n)^2}$
 
@PeterTamaroff There is no issue of convergence: All but finitely many terms are zero.
 
Then $(H,d)$ is a metric space.
 
3:34 AM
It doesn't bound the counterexamples with or without the !. You need to claim that $\lnot P(x)\to x\in\{x_1\ldots x_n\}$.
 
I call it Hilbert space.
@BenjaLim You mean in $S^\infty$?
 
yes.
 
@MarkDominus the $!$ statement would be false if the number of counterexamples is unbounded, whereas (if we add the stipulation that $x_i=x_j\iff i=j$) it holds true if the number of counterexamples is finite. however yours is more compact.
 
@BenjaLim I was just defining the Hilbert space-
 
Dilbert space.
 
3:37 AM
You can then think about the topological space that arises from $H$
And consider $S^\infty$ as a subspace of both the metric and topological space $H$.
@MarkDominus What?
 
$\{x\in S:\neg P(x)\}\in[S]^{<\omega}$
 
I don't like to go around tale taling people, but this user seems to be providing wrong answers everywhere.
@CLarue What is that¿
 
I noticed that. There's also someone whose name starts with "lab" that's similar but not as egregious.
 
He also rolled back an edit that made it look better, that's odd.
 
Yeah, his answers are quite awful. Happily, the site takes care of that automatically.
 
3:41 AM
It's like his answers are algorithmical or something.
 
@PeterTamaroff the notation on the right stands for the set of all sets in $S$ with cardinality less than the exponent.
 
@CLarue And the left?
 
He seems to be one of those folks that writes answers that are so long and without any sort of organizing comments that I'm too lazy to read any of it.
But setting $y'$ equal to a constant I can catch, I guess.
 
PGCIM (Post Graduate Certificate in Industrial Management) from XISS, Ranchi with an aggregate of 70 %.

Bachelors in Electrical Engineering from R.I.T (now N.I.T) Jamshedpur with an aggregate of 68 %.

Class XII from VIDYA BHARATI CHINMAYA VIDYALAYA (CBSE), Jamshedpur with an aggregate of 67 %.

Class X from HILL TOP SCHOOL (ICSE), Jamshedpur with an aggregate of 69 %.
@DylanMoreland =D
 
@PeterTamaroff The set of all elements $x$ of $S$ for which the proposition $P(x)$ fails.
 
3:42 AM
I would like to know who is voting them up. The answer of his that I first noticed as being insane now has +2/0.
 
@MarkDominus Link?
What is all that aggregate stuff?
 
He had 71 rep a while ago.
Now he has 53
Now 51
 
"$17=2^3+3^2$
the sum of the exponents of 2 and 3 is (3+2)
i.e. the sum of the exponents is 5"
 
@PeterTamaroff His posts are rubbish
 
3:45 AM
@BenjaLim I side with you on that.
 
The bottom half of that answer is reasonable, but the top half is rubbish.
 
So are you thinking about $S^\infty$?
 
33 for Generic's answer? wtf!?
 
I didn't upvote I swear!
 
I upvoted it because it was simple and to the point, but it was nowhere near 33 when i did that.
 
3:47 AM
Everytime I feel like giving up on an exercise I put this full screen.
She's like staring right at your soul.
 
I don't think I've seen the full, nonshopped version before
 
Have you seen the "after" picture?
 
@MarkDominus I hate you.
 
Notice also that she is now wearing a niqab.
 
@MarkDominus I'll choose bliss and ignorance.
@BenjaLim
I need some hint in proving $\partial A \cap A^o=\varnothing$
 
3:55 AM
yes
 
I wrote something up there
Basically
 
ys?
 
If $x\in A^o$ then $x\in O\subset A$ with $O$ an open set.
Then $O\cap (X\setminus A)=\varnothing$
 
well yes that has to be true
@PeterTamaroff $\partial A = \bar{A} \cap \overline{X - A}$ YES?
 
I prefer $X\setminus A$
But sure.
 
3:58 AM
$X\setminus(\overline{X\setminus S})~\cap~\overline{S}\cap \overline{X\setminus S}$. Note that the first intersects $\overline{X\setminus S}$ trivially.
 
@anon Should there be an $=$ there?
 
you can put an = sign on the LHS of everything if you want
 
@PeterTamaroff The problem has an easy interpretation
 
@anon I don't understand what you wrote there.
What is $S$?
 
@PeterTamaroff I mean $A$, whatever. It's the intersection of boundary and interior.
 
4:00 AM
@BenjaLim No no
I'm with topological spaces now.
 
Doesn't matter just replace ball with open set
 
@anon Oh, sorry-
 
Every open set about $x \in \partial A$ will contain some point in $A$ or the complement yes?
 
Now if for all open sets $U$ about $x$, $U \cap (X - A) \neq \emptyset$ we have a contradiction yes?
 
4:03 AM
$x\in U$ and $U\cap (X\setminus A)\neq \varnothing$
 
@PeterTamaroff But what does it mean for $x \in \overline{X - A}$?
Every open set $U$ about $x$ will intersect $X- A$ yes?
 
@BenjaLim Yes.
 
In other words there will not exist an open set $U$ about $x$ that is completely contained in $A$ yes?
 
Every nbhd of $x$ will intersect it.
 
In other words $x$ cannot be in $A^{\circ}$ yes?
 
4:04 AM
@BenjaLim Ja.
 
@PeterTamaroff Hence $\partial A \cap A^{\circ} = \emptyset$
@PeterTamaroff helped?
 
@BenjaLim Da.
 
@PeterTamaroff so the problem was easy yes?
@PeterTamaroff Now here's another problem:
 
says the one with the inverse image question
 
We define the closure of $A$ to be the smallest closed set containing $A$
 
4:06 AM
@anon HAHAHHHA mean!
@BenjaLim I know what the closure is!
 
@PeterTamaroff Prove that $x$ is in the closure iff every $U$ open about $x$ intersects $A$
@anon OK OK.
 
:)
 
Ohhhhh man. I'm tired.
 
@anon I wonder if you can tell us your name. Or the initial at least.
@AntonioVargas I want a pony.
 
@PeterTamaroff Tired Santa says "Maybe next year".
 
4:07 AM
bye guys
bye!!
 
nah. I saw DKR recently.
later Benja
 
@BenjaLim cya
 
@BenjaLim Bye Benjamín!
 
'jamin
 
@anon OH MY GOD.
@BenjaLim
You do realize you name contains the word "Jammin" in it?
How awesome is that ?
@AntonioVargas Angry infant kidnaps Rudolph.
 
4:09 AM
@PeterTamaroff The elves go on a manhunt.
You may think the North Pole is some kind of wonderland, but they slaughter 3 or 4 children a year trying to sneak into the compound.
 
@AntonioVargas Stupid elves are stupid. Angry infant lives in Congo. Elves die of malarya and skin cancer.
Remaining group returns to North Pole.
 
@PeterTamaroff First Blood & Diapers
The elves build a funeral pyre for their fallen comrades, paint all of the toys black.
this is madness.
 
@AntonioVargas Elves worship Mick Jagger.
@AntonioVargas No. This. Is.
 
c-c-c-ombo breaker
 
come at me bro
I'm leaving
Byes
 
4:23 AM
@PeterTamaroff later
Well I guess I'm out too. Cya @anon
 
 
1 hour later…
5:48 AM
@anon: when I read this question, I was thinking of Arturo's method. I guess I need to do more with symmetric polys. I did post a proof of Newton-Girard last year, but that is about it.
 
I also thought of Arturo's when I read it, but decided to go the more general route
 
@anon I always appreciate different methods. That's why I upvoted both.
 
user19161
6:19 AM
Hey @rob! How's the trip?
 
@JasperLoy It went very well. We got into town about 6:30, unpacked, and picked up dinner.
 
user19161
@PeterTamaroff You can always tell yourself that the nice picture is another, imaginary person. So that it is not the original person. So there is bliss and knowledge, not bliss and ignorance. That's why when I put up a pic of myself, I always use the real thing, no photoshopping.
 
user19161
6:33 AM
@BenjaLim Of course, that definition only makes sense because it refers to the intersection of all closed sets containing A which is again closed.
 
7:50 AM
Blub.
 
blub?
 
inblublibly.
 
Yep. @OldJohn Just saw your e-mail :-).
 
@JonasTeuwen OK
 
@OldJohn Yes, same Federer. I will reply soon.
 
7:56 AM
@JonasTeuwen OK
 
user19161
@JonasTeuwen Herbert Federer, my idol?
 
Not sure? I mean the writer of phone books.
 
user19161
So did you sleep well last night @jonas?
 
@OldJohn is your gravatar a newton's method fractal?
 
@JasperLoy No. Like 4h.
 
8:00 AM
@DavidWheeler Yep - the three colours represent which of the 3 roots of $z^3=1$ a given starting value converges to (I think)
 
user19161
@JonasTeuwen Phone books! Then yes!
 
@JasperLoy A bit like Doob in that respect?
 
Excellent.
Unwantedly I've got into a discussion about non-commutativity in quantum mechanics with a professor. And he is much better in it... apparently 8-).
 
user19161
@OldJohn Maybe. I saw Doob's measure theory book once. But Federer is quite unbeatable in conciseness! Well, maybe I can beat him, but we'll have to wait another decade or so for any of my books to be published...
 
@JasperLoy You want to write a book nobody reads? 8-).
 
8:04 AM
@JasperLoy I was thinking of Doob's book on Potential theory
 
user19161
@OldJohn Oh I have not seen that.
 
@JasperLoy 846 pages ...
 
user19161
@JonasTeuwen Hmm, they might become the new Bourbaki! Just wait and see...
 
@JasperLoy So five books nobody reads?
 
user19161
@JonasTeuwen Five? I will write three according to my current plans. And Bourbaki has more books than that actually. Actually, it depends on how you count the Bourbaki ones.
 
8:06 AM
Yeah, I know...
 
@JasperLoy if you write better books than Bourbaki, i'll buy first editions :P
 
user19161
Speaking of Bourbaki, I am really not sure how many books there are now. There are all kinds of info on the internet. In fact I remember I was quite confused when I checked out Springer website itself.
 
user19161
And you know what? Amazon seems to have more updated info than Springer sometimes. For example, Lee's Smooth Manifolds second edition was announced for pre-order first on Amazon and then on Springer.
 
Speaking of Amazon, I recently saw a maths paperback available for £82 (new) or £91 (used) - seemed a bit daft :)
 
8:27 AM
Mm. A good day :-).
 
9:12 AM
Good day here too - 2 new books arrived by post :)
 
Cool, which ones?
Two more things that look nice on the book shelf?
 
Gouvea (p-adic numbers) and Cassels (Geometry of numbers)
@JonasTeuwen Might have to think of getting a bigger shelf :)
 
@OldJohn Hmm, I put many of them in my office, I have two big book shelves :-).
 
@JonasTeuwen One day, you will need more
 
@OldJohn But Erdos had very little books... eh?
 
9:18 AM
@JonasTeuwen but a very large brain
 
 
1 hour later…
10:20 AM
good grief - while I have been online this morning, there have been 6 questions from the same person ...
 
oh?
 
I don't see notifications that there are new/changed questions on the main site, nor new comments or changes to questions or answers. I've only noticed this since I restarted my computer after our trip. Does anyone else see this?
 
I still see notifications the same as always
 
The notifications are unreliable for me.
 
@OldJohn Do you know Richard Gill? He was in Cambridge around the time you were there, perhaps a few years earlier.
 
10:30 AM
@JonasTeuwen No - I never came acroos him, I don't think
 
user19161
@OldJohn And also silk underwear.
 
@JonasTeuwen So - he was there at the same time I was. But due to the college system there, I met very few others apart from the 10 students at my college
@JasperLoy ??
 
He is pretty cool and vocal.
 
user19161
@OldJohn Well, he had a skin disease so he has to wear silk underwear and not other kinds.
 
10:33 AM
I was a pretty hopeless student in my undergraduate days :(
@JasperLoy Ah OK!
 
@OldJohn But you were at Cambridge, so hopeless?
 
@JonasTeuwen I might have had some small ability, but I was distracted by many other things going on in my life at the time, and came out with a poor degree - hence the interest in doing a PhD later in life
 
@OldJohn But it ended up quite okay ;-).
 
@JonasTeuwen Yes - I am happy with the eventual outcome :)
 
In an interview Gill stated he didn't really wanted to do a PhD but that it ended up that way.
 
user19161
10:37 AM
@JonasTeuwen Indeed. Life takes mysterious turns.
 
@JonasTeuwen curious
 
Quite a peculiar guy.
 
I've heard claims that, in today's job climate, people take Ph.D.s to avoid having to fight for jobs...
 
Hmm... so they have to fight harder 4 years later? 8-). Postpone fighting! Get Permanent head Damage!
 
@JonasTeuwen Many mathematicians are peculiar (but pure ones more so than statisticians, in my experience)
 
user19161
10:39 AM
@OldJohn And many of them are nutcases.
 
@OldJohn He is quite... mixed in pure/applied. Very pure statistics, so to speak.
 
@JasperLoy Yes - but most of them don't get into the real world much :)
 
user19161
@JonasTeuwen Statistics can be very theoretical, not that I know any.
 
user19161
@OldJohn Because mathematics is more real than $R^3$.
 
@JasperLoy Maybe. Speaking of getting out more, I must get outdoors sometime today
 
user19161
10:42 AM
@OldJohn Yes, walking is the best exercise for the body, according to an ad I saw ten years ago with the model Linda Evangelista.
 
@JasperLoy Agreed - I went hill walking yesterday and really enjoyed the exercise
 
It is very warm outside.
 
user19161
@JonasTeuwen It's worse here bro!
 
Possible. It is very humid here.
 
user19161
Hello @matt!
 
10:44 AM
About 30 deg C. That isn't so bad but the bloody humidity...
 
I don't want to take any oral exams :,(
 
user19161
@JonasTeuwen Yes, it's the humidity you feel. After all, body temperature is 37.
 
user19161
@MattN. Well, try to get it over and done with.
 
Right - time to return to reality for some exercise - back later, folks
 
@MattN. You will do fine. The worst thing is your anxiety, not the exam.
@OldJohn Good luck!
 
user19161
10:47 AM
Oh, he's gone.
 
user19161
@JonasTeuwen You don't need luck for exercise!
 
Then you are not making it hard enough.
The cliché: no pain, no gain certainly applies. Where you can see "pain" in quite some broad context.
 
user19161
I used to go gym once a week. I used to have a Taylor Lautner physique, but not now.
 
Say in handling anxiety for example. Your mind is the boss.
 
user19161
@JonasTeuwen Though in this case, the pain and the gain are not so well-defined.
 
10:49 AM
Yes, Sir. That doesn't matter, the idea is clear.
 
user19161
@JonasTeuwen That's why I need help from that Buddha guy.
 
You don't need his help. Support is what you need!
 
user19161
@JonasTeuwen Thanks Madam.
 
That Buddha guy surely wants you to do it yourself and would support that if you want to, no?
 
user19161
Meow. Now I am also a cat lady.
 
10:51 AM
A crazy cat lady? Have to go to a MSc thesis defense to ask annoying questions. Be back later.
 
 
2 hours later…
12:29 PM
@Jonas: do you actually know, how can I try to talk to did?
 
unfortunately, the belief one can do everything at the last moment doesn't apply to dentist trips. panics
 
@JonasTeuwen Thanks Jonas.
@Ilya Have you tried changing your name and avatar? Or if that didn't work, creating a new account?
@anon ...?
 
@Matt: you think, otherwise it wouldn't work?
 
@Ilya I don't know but I thought he ignored you. So this seems like the only way.
But he doesn't strike me as someone who wants to talk to people.
bbl
 
@anon for you
 
12:48 PM
@Ilya Go here and invite him to chat?
 
@robjohn a good idea - let me try
 
spam flags/votes for deletion please. Sorry gotta run again.
 
ami
@jon
@JonasTeuwen there?
@JonasTeuwen sent you the email.
 
1:06 PM
@Ilya Who?
 
user19161
@Ilya Do you actually know if Matt is ignoring me? It seems to me so, but I wonder why...
 
user19161
Anyway, never mind. I don't think I did anything bad to him ever. At least not that I know of.
 
user19161
1:32 PM
@mattN. I am not sure where I have upset you in the past. I am sorry if I did, but if you don't respond to this, I will take it that you are ignoring me and I won't talk to you from now.
 
does apostols calculus cover as many topics as stewarts?
 
user19161
@ChuckFernández Possibly more and in greater rigour.
 
ok, thanks
 
user19161
@ChuckFernández Don't confuse his Calculus with his Mathematical Analysis. They are different.
 
1:50 PM
what is a good book on PDE?
 
user19161
@ChuckFernández What level are you looking at?
 
noob level
 
user19161
Strauss.
 
is it deep enogh?
 
user19161
@ChuckFernández Why don't you take a look and see if it fits you. Look at the contents and a few pages.
 
user19161
1:55 PM
@ChuckFernández I think if you are reading Apostol's Calculus he treats ODE and PDE there too very well.
 
do you think once i finished Apostols i could read Evans book directly?
 
user19161
@ChuckFernández I think not really. You should at least take some graduate real analysis concurrently.
 
if i read baby rudin would i be ready?
 
user19161
@ChuckFernández Not really.
 
what book should i read before then?
 
user19161
1:58 PM
@ChuckFernández Well, at least some basic measure theory and functional analysis, like big Rudin.
 
what is the name of big rudin?
 
user19161
@ChuckFernández Real and Complex Analysis.
 
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