Mathematics - 2017-06-12 (page 1 of 9)

Adeek

00:41

@arctictern check this nice proof when you come back

1

Q: Proof verification: Zariski topology is quasi-compact

I am solving the following problem from AM commutative algebra. I am solving commutative algebra in my spare time to get good grip in it. Prove that the Zariski topology is Quasi-compact. Here is my proof, I just want to make sure that I have everything correct. We want to prove that X is quas...

@BalarkaSen Yeah if you think about it most human knowledge is really based on analogies. Some discoveries might not be even discovery in the end, but analogy related to something else.

Fargle

I would definitely recommend Surfaces and Essences by Hofstadter and Sander for a more detailed description of knowledge-as-analogy, coming straight from a cognitive scientist horse's mouth.

Adeek

most human thinking is based on Habits and things we do, so I wouldn't be surprised that discovery is just based on analogies.

@Fargle you should read power of habits

it is awesome

Fargle

That's exactly the idea behind the book I just mentioned.

When I have some disposable, I'll check out the book you mentioned. Thanks for the rec.

Faust7

01:01

anyone here alive

id like to take a whack at Topology again (first time i tried it really didnt make any sense ) anyone reccommend what i should have an understanding of / book/s i should understand before giving it a whirl again?

Daminark

I don't know if there's quite a material prereq (you'd need set theory but most topology books start with what you need) so much as a "mathematical maturity" prereq

Faust7

i have a fairly firm grasp on advanced calculus on R or C and a fairly strong understanding of abstract algerbra now just wondering if theres anything else i should understand ( i know last time i really struggled with what a mertric space was)

Fargle

@Faust7 You should have a decent grasp of the properties of open and closed sets in $\Bbb R^n$. Those properties (e.g. stuff about unions and intersections, complements, and so on) directly motivate the topological definitions.

As long as you have that, I'd say you should be fine.

Faust7

ok that stuff is rather easy for me now do i need to understand differential equations?

Fargle

Oh, God, no.

Faust7

01:09

Despite have taken several advanced courses and doing rather well in them i have no idea how any of that crap works

ok maybe ill give it anthor try

any good book for somoene whos particularly slow that you reccomend?

Daminark

Munkres is good

Fargle

I don't know many topology books. Munkres' book is The Topology Bible, though.

Faust7

i know my libary has that ill give it anthor try

Daminark

That and Willard are the two main ones

Faust7

willard k

Daminark

01:11

But I'd say a good way into topology is to understand things well on metric spaces first

Fargle

I found his explanations very enlightening, and his pictures are a good schematic for the types of pictures you should draw as you do topology.

Faust7

yeah and im not entirely sure what a metric space is...

Fargle

@Faust7 A metric space is just a set with some notion of "distance".

Daminark

It's tricky at the onset but it's a question of being presented it right

A metric space is a set $X$ and a function $d:X^2\to\mathbb{R}$, takes in two points and spits out the distance between them

Faust7

whats the diffrence between a vector space and a metric space?

Fargle

01:13

That is, for any two points, you have some non-negative real number associated to those two points which represents the distance between them.

Daminark

Oh they're very different, a vector space is when you have a set so that you can add things together and scale them by numbers

Leaky Nun

@Secret I hope you made some typo in the first two lines of your definition of Ordinal tetration numbers...

Fargle

@Faust7 Vector spaces contain vectors, and you can add them and scale them. A general metric space can have any objects, and you might not be able to add or scale them.

Daminark

Or perhaps better said, "numbers"

Faust7

dam was hoping to relate them to something i understood

Daminark

01:14

Well... There is an intersection between them

Fargle

@Faust7 Well, $\Bbb R$ is a metric space.

Leaky Nun

@Secret it isn't the first time you try to "define" things just to get around some problems, without thinking about whether the definition is valid

Fargle

The distance between $x$ and $y$ is just $d(x,y) = |x - y|$.

Faust7

ic

Fargle

The thing about metric spaces is that you require that "distance" make some sense, in a way that corresponds to the way distance normally works in the real world.

Daminark

01:14

If you have a vector space, you also have to specify what's called a field, which is those objects you use to scale. If that is $\mathbb{R}$, then you can put a norm on the vector space. You can always look at a norm as the distance between a vector and 0

Faust7

there must be a subtley in the definition that broadens the parameteres to include many other things than just R then

Fargle

First, the distance between two different points is always positive, never 0.

Faust7

yeah i actually sort of understand vector spaces form differential geometry

Fargle

(And the distance from a point to itself is always 0.)

Second, the distance between $x$ and $y$ is the same as the distance between $y$ and $x$.

(Duh, right? But that's important.)

Faust7

do you use the way of defining the distance to make sure its always positive?

Fargle

01:16

And third, you want this wonky statement to be true:

@Faust7 Yes.

Daminark

Yeah, when Fargle gives the third statement, you'll see that this implies that distance is positive

Fargle

"The distance from $x$ to $z$ is always less than or equal to the distance from $x$ to some other point $y$ plus the distance from $y$ to $z$."

Daminark

Oh wait you defined it as such :P

Fargle

This is called the "triangle inequality".

To put this way more clearly, if you have a triangle, the length of one side always has to be less than the combined length of the other two, right?

So, like, you couldn't have a triangle with one side of 8, another of 5, and another of 2. It's just not big enough.

Faust7

in euclidean geometry yes...

Fargle

01:18

Right.

And we define "distance" in the abstract so that this is true no matter what.

Faust7

ok

Daminark

Well, in non-Euclidean geometry, the idea is that you have a different metric, so that while the shortest distance between two points isn't a straight line anymore, the law is still true for the metric

Fargle

So, here's a weird distance.

Faust7

i think you can make a triangle with one side that is finite to that are infinite in hyperbolic geometry though...

and the othe 2 are infinite*

Fargle

Let's consider a set {x1,x2,...}.

Faust7

01:21

ok

Fargle

And define the distance between any two different points to be exactly 1.

Faust7

ok

Fargle

This is, in fact, a metric! (Why?)

Faust7

well it follows rule 3

one sec let me find the other 2

yeah it follows all 3 rules

so definitionally its a metric space

the equal part saves you in part 3

Fargle

Indeed it does.

Faust7

01:24

it also makes my head hurt alittle when trying to think about that set

Fargle

This metric is called the "discrete metric".

Leaky Nun

@Fargle so it's countable? ;)

Faust7

is it a trival set? like one with only 2 elements or something?

Leaky Nun

@Faust7 no, any set will do

even $\Bbb R$

Fargle

@Faust7 It can be 2 elements or 20 or as many as the integers or as many as the reals or more.

The discrete metric can be defined on any abstract set.

In fact, this metric is an important example in topology.

There is a (rather boring) class of topological spaces known as discrete spaces.

Faust7

01:25

so the distance isnt eucledian distance its distance in the metric?

Fargle

Right. You ignore Euclidean distance and impose this one instead.

Anyway, back to my belabored point, when one is discussing topological spaces, one question you can ask is whether it's "metrizable", that is, whether there's a metric such that the structure of the space can be described solely by the metric.

Faust7

isnt that kind of like a realtion on the set insteadof a distance?

Fargle

And it turns out that discrete spaces are all metrizable, with the discrete metric as their metric.

Leaky Nun

@Faust7 isn't distance a relation?

Faust7

well yes an anti-syymmetric one

Fargle

01:28

Symmetric, not anti-symmetric.

Leaky Nun

distance is a relation on $S^2 \times \Bbb R$ though.

AKA a function on that.

Fargle

Right. It's actually a function that takes in two points and spits out a real number.

Leaky Nun

AKA distance is a function $S^2 \to \Bbb R$.

Faust7

wait im alittle lost one sec

Fargle

That real number being the distance.

Leaky Nun

01:29

any function on $A \to B$ is a relation on $A \times B$.

@Faust7 are we using the technical term "relation" or just informally?

Fargle

@Faust7 Think of it as being analogous to binary operations from abstract algebra. Those take in two things in the set and give you back a thing in the set.

But now, instead of giving back a thing in the set, the function is giving you their distance, which we represent by a real number.

Faust7

@Fargle which "relation" are you claiming is symmetric

Fargle

@Faust7 Distance. The distance between $x$ and $y$ is the same as between $y$ and $x$. You don't allow negatives.

Faust7

ok i think i understand that part

Fargle

sniped

Faust7

01:31

lol

ok i understand both parts

do we call this something special when we take this distance between two points in a metric space?

like a function or a map or a norm or just call it the distance?

Fargle

The function that does this is usually called the "metric".

Faust7

ok

Fargle

So for example, the "distance formula" is the metric in Euclidean space.

Faust7

i dont think i would of been able to understand that from a textbook

thank you!

Leaky Nun

$d(x,y) = \|x - y\|$

Fargle

01:33

Of course.

Leaky Nun

$x$ and $y$ are vectors.

Fargle

And once you have a notion of distance, you can start to talk about open balls in the metric.

Faust7

so suptract them cordinate wise then square them add and sq root?

Leaky Nun

@Faust7 exactly. $d(x,y) = \sqrt{(x_1-y_1)^2+(x_2-y_2)^2+\cdots+(x_n-y_n)^2}$

Fargle

Let's start with $\Bbb R^2$ with the usual Euclidean metric.

A "ball" (or a disk) in $\Bbb R^2$ centered at some point $x$ is the set of all points whose distance from $x$ is less than some radius $r$.

Faust7

01:35

ok that makes perfect sense so far

Fargle

To put it in symbols, $B_r(x) = \{y : d(x,y) < r\}$.

Faust7

also makes sense

Fargle

This is exactly how we define open balls in an arbitrary metric.

So, let's return to the discrete metric example.

Leaky Nun

@user160069 hi

Fargle

What does the set $B_{1/2}(x)$ look like, where $x$ is some arbitrary point in the set?

user160069

01:37

hi

Leaky Nun

@Faust7 you'll also see $B_r(x)$ as $B(x,r)$ in some text.

Fargle

^

Faust7

a closed? disc of radius 1/2 centred at x

wait

Leaky Nun

@Faust7 notice the sign in the definition

Faust7

open set

Fargle

01:38

Use the definition in symbols I gave, and the definition of the discrete metric.

Leaky Nun

So, in some sense $B$ is a function $S \times \Bbb R \to \mathscr P(S)$

Fargle

Recall that any two different points are at distance 1 from each other.

Leaky Nun

@Faust7 yes

Kasmir Khaan

hi guys in cartesian protuct such as A = (1,0) B = ( 1,2,3) how do I know if we have this ( x,y ) or (x,y,z)

Fargle

So, what points are at distance < 1/2 from $x$?

Leaky Nun

01:38

@KasmirKhaan A=(1,0) is not a set

Kasmir Khaan

2 tuple or 3 tuples ?

Leaky Nun

you meant {1,0} and {1,2,3}?

Kasmir Khaan

@LeakyNun yes

sorry i typed wrong

Leaky Nun

AxB are all 2 tuples

Kasmir Khaan

is it by definition ?

Leaky Nun

01:39

$A\times B = \{(1,1),(1,2),(1,3),(0,1),(0,2),(0,3)\}$

@KasmirKhaan yes. $A \times B := \{(a,b) : a \in A, b \in B\}$

Faust7

uh

Fargle

It's a bit of a trick question, but it's a very important point.

Kasmir Khaan

what if we had A = {x,y,z} and B = { u,v,w}

@LeakyNun Thanks for help btw :)

I don't understand why they all have to be 2-tuples

Akiva Weinberger

$\begin{align}A\times B=\{&(x,u),(x,v),(x,w),\\&(y,u),(y,v),(y,w),\\&(z,u),(z,v),(z,w)\}\end{align}$ @KasmirKhaan

Leaky Nun

@KasmirKhaan $A \times B = \{(x,u),(x,v),(x,w),(y,u),(y,v),(y,w),(z,u),(z,v),(z,w)\}$

@KasmirKhaan read the definition again.

Fargle

01:41

If every distance is defined to be 1 (except the distance between $x$ and $x$, which is always $0$ in any metric), what points are distance $< 1/2$ from $x$?

Leaky Nun

1 min ago, by Leaky Nun

@KasmirKhaan yes. $A \times B := \{(a,b) : a \in A, b \in B\}$

Kasmir Khaan

ahh so we pair 1 element of set A with one element of set B

Faust7

i want to write something like the set $ \for all y in \mathbb {R^2} s.t \{ y \in |x-1/2|

Kasmir Khaan

okay thanks alot! :D

Faust7

wait what

only x

Akiva Weinberger

01:42

Note that $A\times B\ne B\times A$ in general @KasmirKhaan

though they'll always have the same number of elements

Fargle

@Faust7 If we were working with the usual metric, you'd be right (barring some symbology).

But I'm talking about the discrete metric, the weird example I gave, on some arbitrary set.

Kasmir Khaan

@AkivaWeinberger I see , its kinda new topic for me :) thanks again all :)

Faust7

only x would be in the the set

Fargle

Exactly!

And what would, say, $B_{3/2}(x)$ be?

Faust7

cause everything other than itself would be more than a distance of 1/2 away

everything would be in the set

Leaky Nun

01:43

@Faust7 precisely

Fargle

Indeed!

The discrete metric is weird, because every "open ball" is either one thing or everything.

Faust7

Very intresting

Leaky Nun

@Faust7 which gives us the question: what is the topology defined by the discrete metric?

Faust7

May i ask whats the motivation behind this line of thinking where everything is a set distance away?

hmm

Leaky Nun

@Faust7 as a minimalist example of metric

Faust7

01:45

ic

Fargle

@Faust7 Partly, to show that abstract metrics can be entirely non-intuitive.

Leaky Nun

and also because of what I'm going to show you

Faust7

what do mean by the topology? the shape?

the object?

Leaky Nun

@Faust7 are you familiar with topological spaces?

Faust7

no..

Leaky Nun

01:47

never mind then

Faust7

lol

to wikipedia ^^

hmm seems im somewhat familiar with there structure, god math is intresting.

Daminark

Basically, so you know how if you have a metric, you can start talking about open sets? Even if you don't have a metric, you can still grab a bunch of sets and call them open

Faust7

yes

we spent alot of time on odd things that can happen with open and closed sets and the concepts of continuity etc in various calculus courses and analysis

Daminark

That's a topological space, give me a set and call some subsets open. Now, there are some requirements, and they mirror what's true for metric spaces (or normed ones, as you're more used to dealing with)

Faust7

ic

Daminark

01:52

Any union or finite intersection of open sets is open

Faust7

an infinite intersection of open sets is closed etc

Leaky Nun

@Faust7 not necessarily.

Dodsy

hey

Faust7

sorry

Leaky Nun

There is no requirement on what the infinite intersection of open set be.

As an example, consider the sets $\Bbb R - \{r\}$ where $r$ is a rational number.

Faust7

01:54

and infinite intercetion of open substes of a given set is close

Leaky Nun

All those sets are open, and their intersection is $\Bbb R - \Bbb Q$ which is neither open or closed.

Daminark

Yeah, some specific topologies might have this fact, but not all.

Faust7

closed*

ic

Daminark

All you know a priori is that a finite intersection of open sets is open, and a union of open sets is open

Faust7

ic

Dodsy

01:55

ic ic said the blind man

:D

ever heard that riddle

Faust7

it seems like an attempt to find a more flexible way of thinking about a problem

Daminark

@Dodsy my dad says that all the time

Dodsy

Haha, mine too.

Zee

You know your a mathematician when you talk about balls with a straight face

Dodsy

@Zee hairy balls

@Daminark It also took me an abnormally long time to figure out Euler=oiler.

Zee

01:57

lol I remember talking to a geomtry phd student, he was smiling the whole time he was talking about the theorem

Daminark

So the idea behind this generality is similar to most. You can prove a bunch of things in R^n, and you ask how much of the structure is needed to prove certain claims. If you don't need much, and then find you get a nice bunch of sets which also satisfy it, then :)

Faust7

i find mathematics at least for me has been wrought with epiphanies that what was previously taken as fact; was at the very least a convenient misunderstanding or worst blatantly wrong.

Dodsy

I was fishing the other day

and brought up the hairy ball theorem

and my dad didn't believe me

Leaky Nun

@Secret $\omega[4]\omega^+ = \omega[3]^\omega(\omega[3]\omega) = \sup(\omega[3]^n(\omega[3]\omega)) = \sup(\omega[3]^{n^+}\omega) = \omega[3]^\omega\omega = \varepsilon_0$

Zee

Lololol your dad sounds awesome

Leaky Nun

01:59

@Faust7 it isn't too late to change.

Dodsy

Dad?

is that you?

Daminark

@Zee I'm pretty sure I got used to the math meaning of balls before noting that it had its... Other, connotations

Faust7

@LeakyNun It doesnt ever go away?

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$Mathematics$ Mathematics