Mathematics - 2013-03-26 (page 1 of 3)

Charlie

00:16

@jasper Hi!!!!!

DAmn iiit!

Charlie

01:21

@Karl'sstudents :)

Amzoti

Are there any places to learn how to use the MSE search capabilities better? I recall things I had seen before, but nothing I have tried seems to find them (words, MathJax, LaTex, multiple phrases...). What are the tricks to improve searching? Thx!

Charlie

01:39

ZzzzzzzzzzzZzZZZzZZ

hhh

01:58

I am trying to identify this picture, does it have any similar fractal representation?

Karl's students

@Charlie: I have a funny video: youtube.com/watch?feature=fvwp

Charlie

02:24

@Karl'sstudents oh!

Gustavo Bandeira

0

Q: Is my question still too bad?

I made this question some days ago, I realized it was poorly written and then I edited it - at the time, I thought it was because I failed to define "result", now I think I added a better definiton for "result" but I still received no feedback (the question is still closed). The question is kind...

Charlie

@GustavoBandeira tá meio estranha mesmo

Gustavo Bandeira

Sim.

Vô explicar pra eles que sô nordestino.

Charlie

@GustavoBandeira HAHAHAHAHHAHA

@JasperLoy Good night

@Karl'sstudents Good night, see you!

@gustavo Boa noite!

Gustavo Bandeira

@Charlie Jávai?

Eu ia estender a piada. =/

Charlie

02:32

@GustavoBandeira já...tô com soninho.....

@GustavoBandeira Tchau

Gustavo Bandeira

@Charlie Olha

Charlie

@GustavoBandeira oq?

Gustavo Bandeira

@Charlie Aê

Charlie

@GustavoBandeira HAHAHAHAHHAHAHAHAAHHAHAHHAHAHAHAHAH

@GustavoBandeira deixe-me dormir, boa noite!

Gustavo Bandeira

@Charlie Oky. Boa noite.

Charlie

02:37

:)

amWhy

03:04

Hello everyone! anyone?! ;-)

κρανίοπεριπολία

hi

amWhy

I'm not having a good night of it (my time) on main - seems rather slow - so I thought I'd take a little break :-)

κρανίοπεριπολία

having a break is cool

:-D

amWhy

@κρανίοπεριπολία Yes, indeed. Hello, anon! Just saw you "walk in"...

κρανίοπεριπολία

Hi @seaturtles @anon

anon

03:12

@AlexanderGruber @PeterTamaroff Either of you might find my newest expository answer interesting.

hello

not sure how on-topic it was though

κρανίοπεριπολία

(removed)

amWhy

@anon I'll have to read it; you write good expository answers in general!

caveman

03:34

hello

caveman

03:52

@amWhy

amWhy

@caveman hello!!

caveman

good day

hows it going amwhy

amWhy

@caveman so, so...could be better (big sigh)...but could be worse (big sigh)! ;-) How about you?

caveman

what's wrong?

amWhy

@caveman Nothing...well, most everything (!), but I haven't given up yet. One of these days I'd like to reply without qualifications: I'm doing GREAT!

@caveman How are classes going?

caveman

03:56

I finished my classes

I'm sad becauseI am lonely

amWhy

@caveman I always felt sort of deflated at the end of classes, sort of empty...but you're experiencing only a transitional state, and that makes picking up again next term all the more enjoyable...

caveman

yeah, when it ended I felt like nuclear bomb fallout

there is no next term for me

amWhy

And it can get lonely when you love math, but aren't in an academic setting...

@caveman Congratulations! Graduation? What are your plans?

caveman

I suppose I should do a phd

assuming I pass

amWhy

@caveman Go for it! Why not? (and you'll pass)...

On the light side: take a look at this recent post

caveman

04:05

hehe the question is asked by kim taeyun

korean pop singer

amWhy

Interesting...pseudo-name, you think?

caveman

yeah

amWhy

We had a Lindsay Lohan OP, too...sp?

caveman

@amWhy, I guess I will try to, because there's nothing else I know about that I'd like to do - if I do fail I will really have to reevaluate things

I mean I never enjoyed being in university, I never made any friends there - but at least I learned some math

amWhy

@caveman I understand completely. I'd be lost without actively learning...it's like oxygen to me...

caveman

04:11

yeah

amWhy

I know when I first graduated from university (which I procrastinated by earning 4 majors!), I could not really celebrate graduation...All I saw was an end...an abyss, even though I was accepted into a phd program, it still felt like like a huge void.

caveman

I didnt know you could do 4 majors.. wow

I just need to focus on revision and passin my classes

I really feel like it's hopeless because they're all far too difficult but I'm going to put everything into it anyway

caveman

04:36

:(

Ethan

if f(x) is continuous and bounded as x->a, will its derivative also be?

can someone construct a counter example??

anon

sqrt|1-x^2| at x=+/-1

Ethan

thx

caveman

continuous doesn't imply differentiable

Ethan

yes but the converse does

Gustavo Bandeira

05:22

@Ethan Good night.

@caveman You can, but perhaps focusing is better.

@caveman Do it!

Dominic Michaelis

@ethan take x^2 sin(1/x^2) for x-> 0

@anon is you example really differentiable ? it's to early in the morning for me

Ethan

Are ramanujan's sums good for anything?

anon

@DominicMichaelis obviously not (neither is x^2 sin(1/x^2) at x=0 I don't think)

@Ethan entertainment, fun papers, exotic expansions of arithmetic functions

Dominic Michaelis

x^2 sin(1/x^2) is differentiable at 0 (if you define f(0)=0)

Ethan

They look cool, but I don't see how they would be useful for proving or studying other things

@anon what do you mean by fun papers

Alex Youcis

05:36

@Ethan Some or Ramanajuan's work ahs some particularly deep connections to other parts of math. For example, Ramanjauan's conjecture (see here en.wikipedia.org/wiki/Ramanujan%E2%80%93Petersson_conjecture) was not proved until Deligne's proof of the Weil conjectures (one of the most celebrated and deep theorems algebraic geometry has produced)

Ethan

I was just asking about his sums of primitive roots

I can't understand that =/

Alex Youcis

It doesn't matter. I am sorry I misread your question, I was just trying to illustrate that some of the Ramanajuan's most computational out-of-nowhere results have some deep connections to other more "important" parts of mathematics

caveman

@Ethan, yeah they do get used - I saw it in one of my classes but I dont have the example with me

Ethan

what class

caveman

analytic number theory

Alex Youcis

05:40

@caveman What are your mathematical interests?

Ethan

cave math

3

caveman

numbers

Alex Youcis

@caveman That's a classic. I think I might take a class on large numbers next term, but it has small numbers as a prerequisite and I haven't yet taken that :S

Ethan

lol

Dominic Michaelis

real men don't need small numbers

Gustavo Bandeira

05:47

@DominicMichaelis Some of them need it to measure some particularities...

Alex Youcis

@DominicMichaelis Fine, give me 1 then

Dominic Michaelis

1 ist the neutral element of multiplication not a number :D

Gustavo Bandeira

@TobiasKildetoft Same with my questions.

Dominic Michaelis

a large number ? ack(g64,g64)

Alex Youcis

a larger number? ack(g64,g64)+1

Dominic Michaelis

05:50

aleph 1

Alex Youcis

aleph 2

Dominic Michaelis

aleph 3 (whats that one ?)

caveman

whats aleph 2

Alex Youcis

2^(aleph 1) :)

caveman

assuming the continuum hypothesis

Dominic Michaelis

05:52

aleph 2 is the cardinality of $\mathbb{R}$

Alex Youcis

don't give me that

We're in ZFC+CH

caveman

Continuum hypothesis

In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis, advanced by Georg Cantor in 1878, about the possible sizes of infinite sets. It states: :There is no set whose cardinality is strictly between that of the integers and that of the real numbers. Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900. The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that the hypothesis can neither be disproved nor be proved using the axioms of Zermelo–Fraenkel set theory, the standard f...

Dominic Michaelis

as $2^\mathbb{N} \cong \mathbb{R}$ mine is true too isn't it ?

caveman

@DominicMichaelis, that holds in ZF

Alex Youcis

@caveman PS, it's techincally the GHC that I referenced

Dominic Michaelis

05:55

$2^\mathbb{N} \cong \mathbb{R}$ but it's not necessary that it's cardinality is $\aleph_2$ right ?

Alex Youcis

that's correct

caveman

$$\mathfrak c = |2^{\aleph_0}|$$

$\aleph_1$ is the smallest cardinal bigger than $\aleph_0$

is $\mathfrak c = \aleph_1$?

Dominic Michaelis

whats $\aleph_0$?

Alex Youcis

@caveman What don't you ask Cohen?

caveman

06:11

@DominicMichaelis, $\aleph_0 = |\mathbb N|$

Dominic Michaelis

oh so $\aleph_2=|\mathbb{R}|$ if the continuum hypothesis doesn't hold (and there is exactly one set with a cardinality between those

caveman

no aleph 1

anon

no, the negation of CH does not imply |R| is aleph_2

caveman

aleph_2 would be something like number of functions |R->R|

Dominic Michaelis

so there would be somehting like $\aleph_\frac{1}{2}$ ?

caveman

06:15

haha

anon

@caveman still no, that would beth 2

no, alephs are indexed by ordinals, which have no fractions

these issues were covered in my question here by asaf

(also, see the wikipedia page on ordinals, as it is very instructive)

Gustavo Bandeira

@caveman Is the CH proved through logic?

I've read somewhere that it's a paradox. But yesterday I've read somewhere that it's proved via logic.

But I'm not sure.

caveman

@GustavoBandeira, it can not be proved at all!

but the proof it cannot be proved is logic

Dominic Michaelis

it is proved that it can't be proved, and it is proved it can't be disproved too

Gustavo Bandeira

The text told something like: "It can't be proved nor disproved".

Yes.

Dominic mentioned what I read.

caveman

06:21

@anon, great link

anon

CH is "independent of ZFC" (proved by Cohen)

Gustavo Bandeira

Yes. That's a great question.

Dominic Michaelis

that has to do something with goedels incompleteness theorems which I hardly know

caveman

no it doesnt

Gustavo Bandeira

@anon It seems it can't be proved/disproved in ZFC, right?

Walter

06:24

could someone help me with some linear algebra problems?

i really need help... i'm even willing to pay

Gustavo Bandeira

@Walter XD

@Walter Ask it and let's see if someone knows.

Walter

that's how bad i need help with...

anon

@GustavoBandeira right, that's what independent means

I should ask people online for bitcoins to solve their math problems. always wanted a bitwallet.

Dominic Michaelis

@caveman i thought it would be one of those examples for those

Walter

should i ask here in chat or outside?'

Dominic Michaelis

06:25

is it a long question

anon

either

Dominic Michaelis

or difficult

then in main

Gustavo Bandeira

@Walter I guess you'll have more receptivity in the main.

The main is exposed to more people.

Walter

i don't find them difficult... it's just that they all seem so repetitive

and i don't know if i am understanding it correctly

Gustavo Bandeira

@anon Anon, what can you tell me about the processs of making axioms?

Dominic Michaelis

06:26

@gustavo that doesn't mean you get better answers there

Gustavo Bandeira

Although, I'm not really sure if they're made or discovered. It seems there's a huge philosophical discussion on this.

Walter

Let

.....[ 3 1 4 1 5 9 ]
.....[ 2 6 5 3 5 8 ]
A= [ 9 7 9 3 2 3 ]
.....[ 8 4 6 2 6 4 ]
.....[ 3 3 8 3 2 7 ]

How do I tell if the columns of A form a basis of R^5?

Gustavo Bandeira

@DominicMichaelis Yes. It means there's more people to answer, only.

caveman

@anon, rugatu.com

anon

@GustavoBandeira it requires experience, exploration, and a stomache for really formal thinking. what else do you want to know? not really my area.

Walter

06:27

i've always thought each component was represented by the exponent of the R

am i wrong?

each component ( the number of rows)

Gustavo Bandeira

@anon What's your area?

anon

I am studying a bit of algebra and number theory I guess

Dominic Michaelis

@walter a row reduce yields

$$\left(
\begin{array}{cccccc}
1 & 0 & 0 & 0 & 0 & -\frac{23}{11} \\
0 & 1 & 0 & 0 & 0 & \frac{4}{3} \\
0 & 0 & 1 & 0 & 0 & \frac{106}{33} \\
0 & 0 & 0 & 1 & 0 & -\frac{214}{33} \\
0 & 0 & 0 & 0 & 1 & \frac{50}{33} \\
\end{array}
\right)$$

anon

there are six columns and the space is five dimensional

Walter

right

i understand there are six columns

Dominic Michaelis

06:30

as it has full row rank it is surjective

Walter

but isn't that irrelevant? i thought the number of rows were represented by the exponent number of R

Dominic Michaelis

oh six columns

thats bad cause it can't be minimal

Walter

so since there are 5 components in each column and that matches the five dimensional... so.. don't the columns of A form a basis of R^5?

Dominic Michaelis

the thing is a basis is a minimal generating (i hope thats the english word) system

anon

a basis for a dim-5 space must have precisely 5 vectors. six columns means six vectors.

Walter

06:32

so R^2 has 2 vectors (2 columns)?

Dominic Michaelis

as you have 6 columns your system will have 6 vectors, but 5 are minimal, so it can't be minimal, the only thing that may be is that it is generated

anon

any basis of R^2 has 2 vectors

Dominic Michaelis

@walter becarefull $\mathbb{R}^2$ has infinite many vectors, but any basis will have only 2

anon

@caveman thanks for the link

caveman

bitcoin is full of sociopaths and scammers, i recommend avoiding it

Walter

06:34

say {v1, v2}, couldn't this be R^3? but it couldn't be if we are talking basis, correct?

anon

makes for a healthy challenge

Gustavo Bandeira

@caveman Haha

So, you're saying me that there's a place in internet that doesn't have sociopaths and scammers?

caveman

I just think there are really ill people in the bitcoin world

user19161

07:34

Sociopath is a strong term. If one calls people rude when they are not, chances are that these are not really sociopaths.

user19161

I find it really amazing that people keep making mountains out of molehills and molehills out of mountains.

anon

no, the hacker underworld is indeed crawling with sociopaths, especially when there's money and convenient things like untraceability involved

user19161

However, if the great anon says something, I will take it more seriously.

Dominic Michaelis

oh i like this question

Jayesh Badwaik

07:53

@anon have you been to darknet or freenet?

anon

no. (also there is no one network called darknet)

κρανίοπεριπολία

But there is one side called the darkside.

Dominic Michaelis

any moderator here? can i closevote to emigrate to mathematica stack exchange ?

Manoj Pandey

08:10

Interesting

Can we have a proof for this?

Dominic Michaelis

yeah we can but if we would have it wouldn't be a conjecture

Manoj Pandey

but its pretty amazing

Dominic Michaelis

thats why it is famous

Manoj Pandey

hmm..

Dominic Michaelis

what are you doing if you told someone he is wrong in a comment and he doesn'T make anything

i already downvoted him here but seems like he don't care

Manoj Pandey

08:26

@DominicMichaelis: are you asking me?

Dominic Michaelis

i ask everyone

Manoj Pandey

okay

Dominic Michaelis

@JayeshBadwaik do you want to hear something funny ?

anon

vixra

Dominic Michaelis

what is vixra?

Gustavo Bandeira

08:28

New version of viagra.

Dominic Michaelis

but vixra sounds like you can do it on your own

Gustavo Bandeira

High quality.

伟轩

09:01

@anon Hey

I have a topology exam tomorrow

anon

yeah I have a differential geometry exam wednesday

伟轩

@anon Good luck with that.

@anon Thank god there is no measure theory in the midsem

anon

I will pull an all nighter studying since I haven't paid attention all quarter long :/

伟轩

@anon I generally advise people not to do all nighters

@anon But if you really need to then do it

@anon Man no matter how much I study I still feel it's not enough

anon

you're way up there man

伟轩

09:10

@anon What do you mean?

anon

you already have a plethora of cool math under your belt

伟轩

@anon Yeah but that does not mean that I don't fuck up the basic shit

Like I honestly forgot a big chunk of general topology till like last week

anon

I suppose

伟轩

because I haven't looked at it in a while

@anon I look up to you as a senior :)

anon

I am older in age, true

伟轩

09:13

@anon But you know a lot of math

a lot more than I do

I respect you for that.

@anon Right, I should go write up my cheat sheet.

Night @anon

anon

night

Dominic Michaelis

10:01

wtf? this question got an upvote 1 minute after it was posted, they can't tell me they can read that fast

Jayesh Badwaik

10:14

@DominicMichaelis Yes.

Peter Tamaroff

@anon i read it all. I stopped understanding some stuff towards the end of ir =)

Dominic Michaelis

@JayeshBadwaik the first time i read wlog i thought the author meant the lambert w function (which is called sometimes polylog)

Gustavo Bandeira

@DominicMichaelis People vote for friendship.

Dominic Michaelis

11:24

my reputation is a palindrome why are you all so quiet ?

κρανίοπεριπολία

meditation?

Vrouvrou

hi

help me please

0

Q: Question on measurability of multifunction*

If $(T,\mathcal{A})$ is a measurable space , $X$ a meatrizable separable space. $F$ a multifunction from $T$ to compacte subsets of $X$. We want to prove that $F$ is measurable if $\forall C\in X$ ,$C$ closed,$F^{-1}_+(C)=\lbrace t\in T; F(t)\cap C\neq \emptyset\rbrace \in \mathcal{A}$ In my...

homework measure-theory

Mathematician

Can you tell me where the function Arg(e^{i\pi z}) is continuous, i know that Argz is discontinuous for z in the negative axis,

Is it discontinuous for z such that cos \pi z is in the negative axis

?

Dominic Michaelis

whats your definition of the the arg function ?

Mathematician

11:40

principal

(-\pi,\pi]

@DominicMichaelis principal argument

Jayesh Badwaik

11:55

@DominicMichaelis I too thought something similar, that it was some kind of log function.

Gustavo Bandeira

What would be the meaning of topology in: "Learning algorithms based on backwards propagation of errors can be used to find optimal weights for given topology of the network and input-output pairs."?

Is it somehow related to form?

Jayesh Badwaik

12:16

@GustavoBandeira Network Topology

Gustavo Bandeira

Yes, I've seen it too.

Jayesh Badwaik

Then, yes, it is topology in sense of graphs and discrete mathematics.

The word topology in network topology is used in connection with describing the properties of the network that do not depend on the physical parameters like the distance between the nodes (which may lead to loss of information due to attenuation in channels for example), but only on the interconnections. There can be cases where the attenuation is so low that it can be completely ignored, and then it may not matter.

Gustavo Bandeira

12:37

@JayeshBadwaik Got it. Thanks.

But in a more general sense, is it somehow related to form?

Mathematician

@GustavoBandeira can you answer my question?

Gustavo Bandeira

Can I freely associate the word "topology" with some kind of form?

Mathematician

or

Gustavo Bandeira

@Mathematician Last one?

Mathematician

@JayeshBadwaik can u?

yes

Gustavo Bandeira

12:39

Nope. I still don't know CA.

What's the problem with the answer you already have?

Mathematician

ok

i am not sure if it is correct, and if it is correct, what are the possible values of z

Tobias Kildetoft

@GustavoBandeira topology is somewhat related to form, but not quite

Mathematician

since we take the composition, we have to restrict our domain such that e^{i \pi z} doesnot enters the negative axis

Tobias Kildetoft

when we speak of topology, we tend to allow a fairly large amount of deformation without getting something new

Gustavo Bandeira

@TobiasKildetoft Until now, I kinda feel that it's about a form without a form, as Jayesh described: "describing the properties of the network that do not depend on the physical parameters like the distance between the nodes".

@Mathematician Didn't he said that the value of $z$ doesn't exist?

Tobias Kildetoft

12:42

@GustavoBandeira right, we don't care how far there is between nodes

we really only care about how the various nodes are connected

Gustavo Bandeira

Got it.

So, what would be something "new" in topology?

Mathematician

@GustavoBandeira who ?

Tobias Kildetoft

by "new" I meant "different from what we had"

Gustavo Bandeira

Well, the problem is that I conclude that $f$ must not exist... — Twiceler yesterday

Dominic Michaelis

as arg(e^(i pi z))= pi z mod 2 pi you can think what the discontinuouties are

Tobias Kildetoft

12:43

so something that does not change the topology of a graph would for example be moving a node around without changing what other nodes it connects to

something that would change it could be to add or remove an edge

Dominic Michaelis

where only the real part of z does count

Gustavo Bandeira

@TobiasKildetoft So, the difference in the topology of a polyhedra (for example) would be the difference of how it's edges are connected?

Tobias Kildetoft

@GustavoBandeira depends on the topology

Gustavo Bandeira

Hmn. Now it seems that topology is a selective tool where you can decide to look only the aspect you want. Is this it?

Tobias Kildetoft

purely topologically in the usual topoplogy, any two closed curves are "the same" (homeomorphic)

@GustavoBandeira technically, doing topology on something means "pick a topology" and then "look at things up to homeomorphism/homotopy/isotopoy/whatever"

Gustavo Bandeira

12:46

(I'm just naively probing the subject, I've never studied it, just have a naive pre conception).

Tobias Kildetoft

general topology is a strange beast

Gustavo Bandeira

Oh, got it.

One more thing: What mathematical objects have topology? Only geometrical ones? I read that there are open/closed/clopen sets, which I supose that have to do with sets of numbers too (for example).

But until now, most of what I've seen seemed to be linked to geometrical objects.

Altough "geometrical" is meaningless to me. I could only define it as something "visual".

Meaningless because I haven't defined it yet.

Tobias Kildetoft

@GustavoBandeira we can define a topology on any set

a topology is just a way to say "these are the open sets"

usually, we then want that topology to actually have something to do with whatever extra structure we have on the set

Gustavo Bandeira

I guess I understand.

What is required for studying topology?

Oh, and thanks by the math chat btw.

Tobias Kildetoft

for topology, just a general mathematical maturity corresponding to about a year or 2 of university math

so that you have seen the concrete examples that motivate the theory and have gotten used to the type of arguments used

Gustavo Bandeira

13:00

Well. I guess I'll have to skip for now. I'm still on calculus and linalg.

Tobias Kildetoft

@GustavoBandeira topology (for one thing) gives a more general version of what it means for a sequence to converge and what it means for a function to be continuous

(differentiability requires a bit more than just topology)

Gustavo Bandeira

Oh. Then it's completely different of what I was thinking.

Tobias Kildetoft

what were you thinking then?

Gustavo Bandeira

That it was something geometrical.

A friend told me that the video about the sphere inside out is about geometric topology.

Tobias Kildetoft

geometry deals in part with topology

in some sense topology takes part of the geometrical information and forgets the rest

for example, think of a surface in 3d (so like a piece of paper that has been bend or something like that)

if we start with a flat piece and start bending it, then geometrically, we are changing it, since some parts of it are moving closer together

but no matter how must we bend it, it will not change when seen from the point of view of topology (as long as we do not tear it)

Gustavo Bandeira

13:14

Yes, there's a rubber analogy with topology.

Tobias Kildetoft

right

to a topologist, a coffee cup and a doughnut are the same (assuming the cup has a handle)

Gustavo Bandeira

The point I don't get in this analogy is: why the hole makes it different?

I presume that the presence of this hole is some kind of important mathematical pattern.

Tobias Kildetoft

right, essentially, the only thing that one can see topologically is the number of holes

(this is a very loose interpretation of course)

Gustavo Bandeira

Yes. I understand.

Tobias Kildetoft

the way topology works is as I said to define the open sets

on the real line, the open sets are the open intervals and unions of such

(in the topology one usually considers)

do you know what it means for a sequence to converge?

Gustavo Bandeira

13:28

@TobiasKildetoft I guess so.

I've watched a video in which the guy says that only the method of long division teaches the concept of convergence.

Tobias Kildetoft

@GustavoBandeira ok, it is a nice exercise to show that a sequence $a_n$ converges if and only if for any open set $A$ in the reals, there is some natural number $N$ such that for all $n\geq N$ you have $a_n\in A$

(remember that the open sets in the reals are unions of open intervals)

Gustavo Bandeira

He also gives an example (which I don't remember) and it was something like: $x+\sqrt{2}{x} + \sqrt{3}{x}...$

Tobias Kildetoft

the reason this is a nice exercise is that it shows convergence is purely something to do with open sets

Gustavo Bandeira

I'm trying to assimilate unions of open intervals.

I guess I got it.

Now I'm feeling it really have more to do with set theory than with geometry.

Tobias Kildetoft

woops, I formulated the above slightly wrong

we want to say that the sequence converges to some real number $a$ if for any open set $A$ in the reals which contains $a$,...

(otherwise, we would never have any sequences converge)

anon

13:50

@GustavoBandeira For metric spaces open sets can be thought of as those sets with "wiggle-room everywhere" in the sense that $U$ is open iff every point $x\in U$ has an intermediate-lying open ball $x\in B(x,r)\subseteq U$. Balls are a form of room that a point may "wiggle around" in. More generally, though, any collection of subsets generates a topology as a subbase, so it is quite arbitrary (and so "set-theoretic").

Open sets can be thought of as axiomatizing "semidecidable properties" (and properties on a set X are just distinguished subsets...); see Qchu's answer here.

Ultimately though open sets provide a simple, fundamental and unified way of talking about a host of other important topological notions like closedness, continuity, convergence, compactness, homeomorphisms and homotopy, etc. and this is why their definition is standardized as it is.

Gustavo Bandeira

@anon I'm getting it.

Vaguely, because I'm a noob.

Erick

ABCD is a trapezoid with AB<CD and AB parallel to CD. Γ is a circle inscribed in ABCD, such that Γ is tangent to all four sides. If AD=BC=25 and the area of ABCD is 600, what is the radius of Γ?

can anyone help?

Dominic Michaelis

14:51

7777 reputation :D

Charlie

@DominicMichaelis :D

@DominicMichaelis How are you Dominic?

Dominic Michaelis

i am fine

Charlie

@DominicMichaelis Good! a few days from your birthday

Dominic Michaelis

oh how you konw ?

Charlie

@DominicMichaelis ;)

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$Mathematics$ Mathematics