Mathematics - 2012-08-02 (page 5 of 6)

Peter Tamaroff

21:02

@skullpatrol Who are you talking about?

skullpatrol

@PeterTamaroff George Bergman.

Peter Tamaroff

@skullpatrol Does he participate in SE?

skullpatrol

@PeterTamaroff No.

Peter Tamaroff

@t.b. Hey t.b.

t.b.

@Ilya I'm not quite in the mood for measure theory...

And given that you don't seem to have time for this

@skullpatrol thanks.

@PeterTamaroff Hey, Peter

@Ilya are you still there?

skullpatrol

21:10

@t.b. Do you mind if I ask you if you have taught any math to students below first year university?

Ilya

@t.b. I am

@t.b. why?

Peter Tamaroff

@skullpatrol I was going to ask if he was a math instructor. Many people that do research tend to be instructors.

t.b.

very good!

^^ Jasper will love this, just like the good ol' days

@skullpatrol I did teach at high schools during my early years in university but that was a long time ago.

skullpatrol

@PeterTamaroff He is a retired Berkeley Professor.

Ilya

@t.b. ^_^ I visit this chat quite rarely now, so I don't know if there is still a lot of deleted messages

t.b.

21:16

@Ilya I'm not around that often either. Once or twice a week.

skullpatrol

@t.b. Do you remember teaching the left to right order of operations rule?

Ilya

icic

t.b.

@skullpatrol I said high school, not kindergarten or elementary school :)

I never had to teach that and stuff like Bedmas and Pemdas, FOIL and whatever don't exist around here.

Ilya

bbl guys

t.b.

see you Ilya

skullpatrol

21:22

later

Peter Tamaroff

Anyone knows about a book with good topology exercises? Or maybe an open source of exams and stuff?

I did one about the $T_1$ axiom yesterday which Mark D. got from Kelley's book.

skullpatrol

@t.b. Thank you for the comments.

robjohn

@Ilya: sorry to miss you. See you later.

@t.b. how goes the day?

anon

In group theory what are central, regular and free factors? See e.g.http://en.wikipedia.org/wiki/Retract_(group_theory)http://en.wikipedia.org/wiki/Transitively_normal_s‌ubgroup http://en.wik‌ipedia.org/wiki/Central_subgroup

t.b.

@robjohn Oh, don't tell Matt but I hope Summer will finally show up :/

Old John

21:28

@PeterTamaroff Not open source - but I once used the Schaum Outline book on General Topology and it had loads of exercises

robjohn

@anon got me...

t.b.

How are you doing?

robjohn

@t.b. What is the weather like now?

@t.b. Doing better. We were on vacation last week, and got back to the flu for the beginning of this week.

Peter Tamaroff

@OldJohn Oh, cool. I'll take a look. How are you?

Old John

@PeterTamaroff Great, thanks - did you get that other book you wanted (can't remember which it was)?

t.b.

21:30

@robjohn Somewhat warmish. 25 during the day 16 during the night (Celsius). Stormy. Raining every other day. More often than not. We had four or five days of really sunny weather but the rest of the summer was real crap so far.

robjohn

@t.b. Some of the US could use that kind of weather. The middle of the US (the bread basket) has what they call "sizzling" weather. Over half of the counties in the US have what are classified as drought conditions.

Peter Tamaroff

@OldJohn Willard? I did.

Old John

@robjohn darn - summer here has been a total washout - more rain than I can remember :(

robjohn

@OldJohn I've seen some of the weather that London has been having. I don't know how similar that is to yours.

Old John

@PeterTamaroff Great - Schaum has many more exercises which are more straightforward - I think it is a much under-rated book

@robjohn Ours is like London - but somewhat wetter - we are on the west of the UK and it is always a bit wetter here

t.b.

21:34

@robjohn Oh, well, I could offer a few hailstorms that destroyed my garden early July.

Would that be acceptable?

robjohn

@PeterTamaroff: you saw the full proof of the formula for the number of factors of $p$ in $n!$ that I posted yesterday?

Peter Tamaroff

@robjohn Nay!

robjohn

@t.b. :-)

@PeterTamaroff: Suppose
$$
n=\sum_{k\ge0}d_kp^k
$$
Then
$$
\left\lfloor\frac{n}{p^j}\right\rfloor=\sum_{k\ge j}d_kp^{k-j}
$$
So the number of factors of $p$ in $n!$ is
$$
\begin{align}
\sum_{j\ge1}\left\lfloor\frac{n}{p^j}\right\rfloor
&=\sum_{j\ge1}\sum_{k\ge j}d_kp^{k-j}\\
&=\sum_{k\ge1}\sum_{1\le j\le k}d_kp^{k-j}\\
&=\sum_{k\ge0}d_k\frac{p^k-1}{p-1}\\
&=\frac{n-\sigma_p(n)}{p-1}
\end{align}
$$

t.b.

ouch...

Bill Dubuque

@skullpatrol Besides Bergman's blurb, see "Order of operations" and other oddities in school mathematics by Hung-Hsi Wu, a Berkeley mathematician who has given much thought to mathematics education.

Peter Tamaroff

21:37

@t.b. Darn! Quite a response.

Old John

@PeterTamaroff If you get the Schaum book, start reading at chapter 5 (only need to look back at the first 4 if there are any definitions you need to check)

Bill Dubuque

@t.b. Hmm... hailstone problems are undecidable in general.

Peter Tamaroff

@t.b. I commented...

t.b.

@BillDubuque yes, but maybe robjohn has some Bézoutka that helps as a countermeasure...

So I hoped.

Peter Tamaroff

@t.b. Am I right in saying that given a set $X$ and its power set $2^x$, $\{\varnothing,X\}$ is the smallest and $2^x$ is the largest topology wrt the partial order of inculsion?

t.b.

21:41

@PeterTamaroff sure

Peter Tamaroff

That is, if $\mathfrak I$ is any topology of $X$, $\{X,\varnothing\}\subset \mathfrak I\subset 2^x$

t.b.

yes, that's right.

Peter Tamaroff

@t.b. I read about some order theory yesterday out of Halmos.

Old John

@PeterTamaroff and the collection of all topologies can be thought of as a lattice

Peter Tamaroff

@OldJohn I couldn't get what a lattice is.

Let me write what I understood.

Say $(X,\leq)$ is a partial order.

skullpatrol

21:43

@BillDubuque Thank you :-)

Old John

@PeterTamaroff OK

t.b.

@PeterTamaroff Think of the unit interval with the usual order.

Or the natural numbers ordered by divisibility.

Bill Dubuque

@t.b. Rob's too busy twisting his mean square into a Mobius strip (see above)

Peter Tamaroff

Then given a subset $E$ of $X$, we say $a\in X$ is a lower bound of $E$ if $a\leq x$ for any $x$ in $E$ and $a'$ is an upper bound if $x\leq a'$ for any $x$ in $E$. Now let $E^*$ be the set of all upper bound of $E$ and let $E_*$ be the set of all lower bounds of $E$. If $E^*$ has a least element, we call it the supremum of $E$. Similarily, if $E_*$ has a last (or greastest) element, we call it an infimum.

A lattice $L$ is a poset such that any pair $\{a,b\}$ of elements of $L$ has both an infimum and a supremum.

Old John

for a lattice, it might help to draw a diagram of a 3 element set and all its subsets - draw a line from a set to a superset - put the sets of various sizes at the same level
then visualise that "lattice diagram" getting bigger as you move to sets at the top with more elements

Peter Tamaroff

21:46

@OldJohn Lattice translates to grid or net in Spanish.

Old John

@PeterTamaroff yes - that is the definition - but I think it is important to get an intuitive feel for what a lattice looks like

robjohn

@t.b. I worry about the limit, too. I have pressed it, perhaps during a second or two :-)

Peter Tamaroff

@OldJohn What I understand is that a lattice somehow has a "neat" order.

skullpatrol

@robjohn Have you finished singing like Chubby Checker?

t.b.

@BillDubuque I see. My favorite Möbius picture still is the one looking like a road and a sign post saying "no parking this side". Couldn't find that one online though.

robjohn

21:49

@BillDubuque Hey!

Old John

@PeterTamaroff yep - but one in which it is not always possible to say which is "bigger" out of two elements

skullpatrol

WOW

Peter Tamaroff

@robjohn Awesome!

Old John

@PeterTamaroff check your email ;)

skullpatrol

Twist it mean square

t.b.

21:51

@PeterTamaroff One you are familiar with is given by the natural numbers and you say $m \lt n$ if $m$ divides $n$.

Then certainly not all elements are comparable.

Peter Tamaroff

@t.b. I see.

@t.b. That not all elements are comparable isn't true for posets already?

robjohn

@skullpatrol Bill was trying to convey that tune earlier.

anon

@PeterTamaroff it's already true for posets, but lattices are a subclass of posets

t.b.

@PeterTamaroff sure, but that's a poset that happens to be a lattice. The lattice property is one that a poset may or may not have. I'm claiming that the natural numbers with respect to this order are a lattice.

Peter Tamaroff

@anon What do you mean by "subclass"?

anon

21:53

every lattice is a poset but not vice-versa

Peter Tamaroff

@t.b. OK, OK.

skullpatrol

@robjohn And I asked you to "twist it" on a Klein bottle long time ago :)

anon

"Not all elements are comparable" - in some posets this is true, in others (linearly ordered sets) it is not true

Peter Tamaroff

@anon linearly=totally, correct?

anon

yes

t.b.

21:55

@PeterTamaroff So can you describe the upper bound and the lower bound of two elements in this example?

robjohn

@skullpatrol the transparency might prove difficult.

t.b.

@robjohn nice :)

Peter Tamaroff

@t.b. Say I pick the pair $\{4,8\}$.

My order is $m \leq n$ whenever $m\mid n$

robjohn

@t.b. This was spawned from the "steps" strip

anon

it's easy to find meets & joins of comparable elements, try something different

Peter Tamaroff

21:56

(It is the weak order of $m$ is a proper divisor of $n$.)

@anon Meets and joins?

t.b.

@PeterTamaroff sup and inf

anon

greatest lower bounds and lowest upper bounds

Peter Tamaroff

@anon Why are they called like that?

I mean meets and joins.

t.b.

later

anon

dunno

Peter Tamaroff

21:57

Ok, say I pick $\{7,13\}$

t.b.

you've got a knack for boring examples :)

anon

try 20,24

t.b.

way better

Peter Tamaroff

@anon Ok, set $E=\{20,24\}$

The supremum is the $\rm lcm$

Right?

And the infimum is the $\gcd$

t.b.

yep

Bill Dubuque

22:01

@robjohn That's really mean and twisted! But you're really much too square to be so twisted.

anon

yep, that's all you need to know

robjohn

4

Higher resolution

skullpatrol

That is why he is the square of mean ):<

Old John

@PeterTamaroff So - that is what a lattice looks like - and the collection of all topologies on a set $X$ is a lattice

Peter Tamaroff

@t.b. Primer are not comparable wrt $\mid$

t.b.

22:02

@PeterTamaroff the reason is in set theory/topology. The power set of a set is ordered by inclusion. Can you see what supremum and infimum of two elements are?

Bill Dubuque

@robjohn That should be your gravatar.

t.b.

@PeterTamaroff yes. It's a pretty complicated order but one you know quite well.

Peter Tamaroff

@t.b. $A \cup B$ is the $\sup$

$A\cap B$ is the $\inf$

Oh! That's why Willard write $\{a,b\}$ and uses $a\wedge b$ and $a\lor b$

t.b.

@PeterTamaroff exactly. So the inf is where the sets meet and the sup is what you get if you join them together.

Peter Tamaroff

@t.b. Heheheh OK!

skullpatrol

22:04

@BillDubuque Opps please refresh your browser...

Peter Tamaroff

This is starting to make sense.

Old John

@PeterTamaroff exactly!

t.b.

@PeterTamaroff with the divisibility order you have a very similar situation, no?

Peter Tamaroff

@OldJohn It has something to do with Boolean algebra?

Old John

@PeterTamaroff very similar

t.b.

22:05

sure

Peter Tamaroff

@t.b. Well, if we see a number as the set of its divisors.

user19161

@robjohn What did you use to draw that?

robjohn

@JasperLoy Mathematica

Peter Tamaroff

If we call $20=\{1,2,4,5,10,20\}$

Old John

@PeterTamaroff do you not consider 20 as a divisor of 20?

@PeterTamaroff :)

Peter Tamaroff

22:06

And say $12=\{1,2,3,4,6,12\}$

skullpatrol

ALL
HAIL
KING
robjohn

user19161

@skullpatrol No, all hail Skull!

Peter Tamaroff

Wait, no. We have to exclude prime divisors?

skullpatrol

@JasperLoy I'm only a hail stone :)

Peter Tamaroff

Because $12\cap 20=\{1,2,4\}=4$

Old John

22:08

@PeterTamaroff - well, that is what you would consider to be "equal" to 4, isn't it?

Peter Tamaroff

@OldJohn OH! DUH!

Old John

but you will hit problems with gcd :(

Peter Tamaroff

@OldJohn Why?

Isn't $\gcd(12,20)=4$?

t.b.

Old John meant lcm

Old John

@PeterTamaroff sorry - I meant the problem will be with lcm

Peter Tamaroff

22:10

$12\cup 20=\{1,2,3,4,5,12,20\}$

Old John

@t.b. thanks! (too much wine here tonight)

Peter Tamaroff

@OldJohn Maybe we should define it in terms of the $\rm lcm$ to make it work?

user19161

@OldJohn What wine are you having? I also wonder what Jonas is drinking now.

Peter Tamaroff

${\rm lcm}(12,20)=60$

Old John

@JasperLoy a rather nice Sicilian red :)

user19161

22:11

@OldJohn I don't really drink but I do prefer red to white.

Peter Tamaroff

@OldJohn That's why maybe I should try with proper divisors.

skullpatrol

@robjohn If you plan to use your mobius strip as a (gr)avatar you'll have to shrink it down a little to fit it all in.

Old John

@JasperLoy Rioja is my favourite

robjohn

@skullpatrol yes, I know it is a bit bigger than 512 across

Old John

@PeterTamaroff Hmm - not sure this analogy is a terribly good one

Peter Tamaroff

22:12

@OldJohn La Rioja is a province in Argentina.

Old John

@PeterTamaroff also in Spain

Peter Tamaroff

@OldJohn Prolly ours got named after them,

skullpatrol

@robjohn I hope you don't mind me testing it out :)

It is just so cool looking.

Old John

@PeterTamaroff Have you ever done an exercise like listing all the possible topologies on a finite set like $\{a,b\}$?

user19161

@skullpatrol Who do you want to shoot with that gun?

Old John

22:15

or $\{a.b,c\}$

user19161

@OldJohn You need another edit.

Bill Dubuque

@robjohn Would it look better with the head upright vs. sideways?

Peter Tamaroff

@OldJohn I have seen it. It shouldn't be terribly hard.

\begin{vmatrix}
{\left\{ {0,1,2} \right\}} & {\left\{ {0,1,2} \right\}} & {\left\{ {0,1,2} \right\}} \cr
{\left\{ {0,1} \right\}} & {\left\{ {1,2} \right\}} & {\left\{ {2,0} \right\}} \cr
{\left\{ 0 \right\}} & {\left\{ 1 \right\}} & {\left\{ 2 \right\}} \cr
\varnothing & \varnothing & \varnothing

\end{vmatrix}

skullpatrol

@JasperLoy Henry T. Horton said he wanted a gun.

Peter Tamaroff

Could someone fix that for me?

Old John

22:16

@PeterTamaroff OK - list them - then draw them in a lattice diagram - it helps

Peter Tamaroff

I want a matrix

@OldJohn Is that a good lattice?

Old John

@PeterTamaroff OK - take that matrix, then merge all the top elements together and the bottom elements

skullpatrol

Hi @HenningMakholm

Peter Tamaroff

@OldJohn Those are topologies!

Old John

i.e. write $\emptyset$ just once at the bottom

and write $\{0,1,2\}$ just once at the top

skullpatrol

22:19

or {}

Old John

and then draw lines to show which are included in each other

Peter Tamaroff

$$\begin{matrix}
{} & {\left\{ {0,1,2} \right\}} & {} \cr
{\left\{ {0,1} \right\}} & {\left\{ {1,2} \right\}} & {\left\{ {2,0} \right\}} \cr
{\left\{ 0 \right\}} & {\left\{ 1 \right\}} & {\left\{ 2 \right\}} \cr
{} & \{ \}& {}
\end{matrix} $$

Old John

@PeterTamaroff Hmm - we are getting a lattice of subsets here, rather than a lattice of topologies, but never mind

Peter Tamaroff

@OldJohn Yes, I know. Wait! =)

robjohn

@BillDubuque If you put the head at 90° you notice the orientation change when it comes around.

skullpatrol

22:22

just like the feet

robjohn

@skullpatrol except I can account for that by using an odd number of feet :-)

Bill Dubuque

@robjohn Right, you need an asymmetric head, maybe winking, or tongue, or ....

Jonas Teuwen

@robjohn Oh... how did you do that? 8-).

@tb Hi hi.

robjohn

@JonasTeuwen Mathematica, of course :-)

t.b.

a duplicate

@JonasTeuwen hi

skullpatrol

22:27

@robjohn Or could you orient them alternatingly?

Bill Dubuque

@robjohn This one has the advantage that the face is easier to perceive by those who have no clue what it's supposed to be.

Jonas Teuwen

Is it a good thing or a bad thing if your advisor doesn't give you a problem but lets you work on whatever you want...?

robjohn

@skullpatrol I could do that and have an odd number

Jonas Teuwen

Many others I know have like a very strict thingie to work on.

t.b.

@JonasTeuwen depends on the student.

Jonas Teuwen

22:28

I suppose.

skullpatrol

@robjohn Just like the feet :)

Jonas Teuwen

@tb Then I suppose it is a good thing in my case. I also have much more freedom in going to whatever conference I like :-).

skullpatrol

@robjohn The downward curving mouth would play the role of the arch in the foot, right?

Jonas Teuwen

I'll have surgery next week to remove the foot from my mouth.

t.b.

@JonasTeuwen It really depends on the student. While I guess that more liberty is generally better for strong students, I think it's an important skill to sit something through till the bitter end. If you have a lot of liberty then there's the danger that you avoid the really hard questions and jump around until you find something that "just does itself". That's very nice but facing difficulties and work through them is a non-negligible skill later on.

Jonas Teuwen

22:32

Well, it actually makes me work on like very hard problems. Not sure if that is good either.

Old John

@JonasTeuwen I found it really useful when at the beginning, my supervisor gave me a definite problem to work on ("here's a paper from 20 years ago - can you make the same sort of ideas work in this more general setting") - and then later gave me more freedom to follow my own ideas

Jonas Teuwen

@OldJohn I did that for my MSc thesis :-). And I completed it in ~3 weeks instead of 10 months. So I suppose he has some belief in that I am able to do it.

Old John

@JonasTeuwen I skipped that step :(

Jonas Teuwen

It is basically compulsory in NL.

Old John

@JonasTeuwen then you are on to stage 2 already - go for it!

Jonas Teuwen

22:34

I do have some results about kernels of semigroups. I am able to replace the variable with something much more general :-).

Old John

@JonasTeuwen good - does it look like it could be made into a paper?

Jonas Teuwen

@OldJohn Certainly!

But I was not allowed to put it on the internet yet my advisor said 8-(. Also, I have some stuff in physics.

Old John

@JonasTeuwen Excellent - then I think your supervisor is doing the right thing :)

Jonas Teuwen

2,5 papers. Also... not yet published.

@OldJohn I guess so. Seems like the best one I could have. I am also am employed by the university and get money from his grant for conferences and he has the highest national one :-).

Old John

@JonasTeuwen sounds like you are in an excellent position :)

Jonas Teuwen

22:37

Well, I don't know the whole construction but some other professors said there was no money, and then he said: "well, I will fix something".

Yes. So more conferences! Helsinki. Plane ticket. Need to buy it now. But I forgot when it was.

Henning Makholm

Sounds like he really has confidence in you and is not just shirking his advisory duties.

Jonas Teuwen

I think so - or I hope so.

Old John

@JonasTeuwen 29-31 August

Jonas Teuwen

@OldJohn Sure. But... maybe it is cancelled already! 8-). And I need to find a hotel.

@OldJohn Ahh, that helped in finding it! Thanks!.

Peter Tamaroff

$\Huge \text{ This might take up some space}$ @OldJohn

$$\eqalign{
& {T_\emptyset } = \left\{ {\emptyset ,X} \right\} \cr
& {T_0} = \left\{ {\emptyset ,\left\{ 0 \right\},X} \right\} \cr
& {T_1} = \left\{ {\emptyset ,\left\{ 1 \right\},X} \right\} \cr
& {T_2} = \left\{ {\emptyset ,\left\{ 2 \right\},X} \right\} \cr
& {T_{01}} = \left\{ {\emptyset ,\left\{ 0 \right\},\left\{ {0,1} \right\},X} \right\} \cr
& {T_{02}} = \left\{ {\emptyset ,\left\{ 0 \right\},\left\{ {0,2} \right\},X} \right\} \cr
& {T_{10}} = \left\{ {\emptyset ,\left\{ 1 \right\},\left\{ {0,1} \right\},X} \right\} \cr

t.b.

22:40

I wonder why this nonsense is still open.

Jonas Teuwen

@PeterTamaroff Oh man. Scary.

Henning Makholm

@JonasTeuwen Opening his secret money caches is a rather more affirmative indication than merely letting you plutter away in an office by yourself.

Henry T. Horton

calculus?!!!?!??!!?

Jonas Teuwen

@HenningMakholm Ah! :-). Yeah, perhaps.

@t.b. Too. little. votes. (added one).

@HenryT.Horton Scary eh.

Old John

@PeterTamaroff OK - now draw a diagram with all the $T_n$ things and show the inclusions between them - you will have a lattice

t.b.

22:42

@JonasTeuwen mine withered away :/

Peter Tamaroff

@OldJohn I'm missing no topology, right?

Jonas Teuwen

8-).

Old John

@PeterTamaroff not sure - not checked them all yet!

Jonas Teuwen

@PeterTamaroff Looks like a mind numbing activity... Shall I give you a five element set?

Peter Tamaroff

@JonasTeuwen Oh, god, NO!

Old John

22:42

@JonasTeuwen NO!!

Henning Makholm

@t.b. General unease with formulas that contain non-matched "d"s? I always think that they may possibly have some formal interpretation that makes perfect sense, which I'm just too uneducated to know.

Peter Tamaroff

One does not simply compute all possible topologies of a 5 element set.

Jonas Teuwen

@PeterTamaroff One time is enough? 8-).

There perhaps are many of those? Radon Nikodym derivatives... differential forms?

Peter Tamaroff

I particularily like this answer @t.b. "fuck moderator fuck moderator fuck moderator fuck moderator fuck moderator"

Jonas Teuwen

Still have to figure out how that goes between pull back and pull forward (RN der and diff form).

Henning Makholm

22:43

@PeterTamaroff Why? There are only 4 billion possible ones to check.

Peter Tamaroff

@HenningMakholm 4 billion? Really?

Jonas Teuwen

The proof is in the pudding.

t.b.

@HenningMakholm I guess so. But it looks more like a case of trolling to me.

Henning Makholm

@PeterTamaroff The size of P(P({1,2,3,4,5})). Some of these are topologies, others are not.

Jonas Teuwen

I think I forgot something, so I am thinking if I forgot to think about something I might forget... hmm.

Peter Tamaroff

22:45

@HenningMakholm Oh, dear!

Jonas Teuwen

Too complex. Need beer.

user19161

@PeterTamaroff OMG!

Jonas Teuwen

Or Finlaggan.

Peter Tamaroff

@JonasTeuwen I think you need whiskey or vodka for that!

Jonas Teuwen

I don't have any whiskey.

Old John

22:46

@JonasTeuwen Finlaggan would be best

Jonas Teuwen

And vodka is for Polish people and Russians.

Henning Makholm

@JonasTeuwen Do you have whiksy?

N3buchadnezzar

@JonasTeuwen Two beers good, four beers bad.

Old John

@JonasTeuwen I thought you had 38 bottles?!

Jonas Teuwen

And crazy Scandinavians.

Peter Tamaroff

22:46

@JonasTeuwen Vodka is for real men.

Jonas Teuwen

@OldJohn Whisky.

Old John

@JonasTeuwen Ah! - misread :)

Jonas Teuwen

@PeterTamaroff Okay. I am not a real man.

user19161

@JonasTeuwen Remember, whisky is the BrE spelling, and whiskey is the AmE or IrE spelling.

N3buchadnezzar

@JonasTeuwen I managed to drink too much beer yesterday, go figure.

Jonas Teuwen

22:46

@JasperLoy No. Wrong.

Peter Tamaroff

@JonasTeuwen You're harmonic real.

Jonas Teuwen

It is not about the spelling, it is about the origin.

@PeterTamaroff Yea...

Peter Tamaroff

@OldJohn I think I'm not missing any topologies.

user19161

@JonasTeuwen Sure, if you think that way. I am just telling you what you would see in a dictionary.

Old John

@PeterTamaroff I think you are right

Jonas Teuwen

22:47

@JasperLoy Well, it would be like saying my name is different in American English...

user19161

@JonasTeuwen You are Batman, and I am Superman.

Jonas Teuwen

@JasperLoy Yep.

So what the bloody heck did I forget?

I'm not even sure if I did forget something.

user19161

@JonasTeuwen No, that is because whisky has entered the English language as a word. Your name has not. It is just a proper noun.

Jonas Teuwen

I locked office... thinking "I must not forget this".

Old John

@PeterTamaroff when you draw them in a lattice diagram, all the ones with very few open sets are at the bottom, the ones at the top have too many open sets

t.b.

22:49

@PeterTamaroff There are 139 homeomorphism types and 6942 distinct topologies.

user19161

@JonasTeuwen You forgot the phone book?

Jonas Teuwen

@JasperLoy If the bottle says... "Scottish Whisky" you would name it differently in another country?

@JasperLoy Ah! That is one of the things I forgot that I forgot!

There is something else.

It involved three words and one was "random". Oh well... Finlaggan.

user19161

@JonasTeuwen The bottle may be labelled as Jonas Whiskey if made in Ireland, but if you use British English, Jonas Whiskey is still whisky.

Peter Tamaroff

@t.b. Wait. I have only 14 topologies for a 3 element set.

Jonas Teuwen

That's an academic issue which is way above my head, @JasperLoy. I'll just drink it. K?

Old John

22:51

@PeterTamaroff don't worry - that is enough to be going on with :)

user19161

@JonasTeuwen You were the one asking about style guides bro!

Peter Tamaroff

I should have 29! I'm really missing something!

user19161

@JonasTeuwen Discrete/continuous random processes?

Jonas Teuwen

@JasperLoy Nope.

@PeterTamaroff Man... don't do that. Mind numbing. You surely understand by now.

Next up: $\sigma$-algebras?

user19161

@PeterTamaroff You are missing ... whisky!

skullpatrol

22:52

Random number generator?

Peter Tamaroff

@JonasTeuwen But it is fun!

user19161

@PeterTamaroff You can spend your time more constructively!

Old John

@PeterTamaroff only do this stuff once in your life - after that it is tedious

Peter Tamaroff

@OldJohn Hehehe true.

Jonas Teuwen

@PeterTamaroff Ah! Right! I always wanted to know the $\sigma$-algebra on $\{1, 2, 3, 4, 5\}$. Can you help me please?

user19161

22:53

@OldJohn Or one can just trim down to fewer elements.

Peter Tamaroff

If I ever teach topology I will give this question to students.

t.b.

@JonasTeuwen drolliger troll

Jonas Teuwen

Finnair/SAS/KLM. Which one is less likely to drop from the sky?

Peter Tamaroff

Write all possible topologies of a 4 element set.

user19161

@PeterTamaroff 3 elements will do.

user19161

22:54

@PeterTamaroff Too many, too many.

Peter Tamaroff

@JasperLoy That's the point!

Old John

You just need a mental image of the topologies on a set - all the ones at the bottom don't have enough open sets to be interesting - the ones at the top have too many open sets - the ones in the middle are fun

Jonas Teuwen

8-).

Henning Makholm

@JonasTeuwen All offer excellent survival chances. Finnair and KLM are likely to have direct flights, though.

Peter Tamaroff

@OldJohn Hehehe OK. But which ones am I missing. I don't get it.

Old John

22:55

@PeterTamaroff don't worry about it

Jonas Teuwen

@HenningMakholm Yeah, SAS only takes like 12 hours.

t.b.

This is my favorite 4-point space

Old John

@PeterTamaroff more useful to think about which ones are $T_1$ or Hausdorff, maybe

Henning Makholm

@JonasTeuwen Don't diss the opportunity of a break in Copenhagen, though.

Jonas Teuwen

@HenningMakholm Oh yeah! They pay anyway.

Coffee Collective is there.

Peter Tamaroff

22:56

@OldJohn I'll try thing bout it.

Jonas Teuwen

So I can surely motivate it: "Needed a coffee break, bro. You understand that right? Also like quality.".

user19161

My favourite set is the empty set.

Old John

@HenningMakholm been meaning to ask: does Denmark have the same sort of odd things as Sweden - like surströmming, snus, ...?

t.b.

@OldJohn there are very many interesting finite T_1 spaces. There might even be an OEIS entry for them :)

Henning Makholm

@OldJohn Surströmming and snus are exclusively Swedish vices. Danes have a thing about pickled herring and NH4Cl flavored candy, but that is more pan-Scandinavian, I think.

Peter Tamaroff

22:58

@OldJohn Found the missing ones!

Old John

@t.b. Ah yes - when I said "interesting" above, I meant "interesting to an analyst" :)

user19161

@OldJohn Then they are mostly just Hausdorff.

Old John

@HenningMakholm I can cope with snus - but not Surströmming - that stuff is evil :(

Transcript for

$Mathematics$ Mathematics