#serious-discussion
long matrix
Ok, next, what is the inverse loop to the one i drew
In this process I am also half convincing you this thing is actually a group 
since I stated but never proved
hidden bough
Ok, next, what is the inverse loop to the one i drew
itself?
long matrix
right
the reversed loop
in fact, take any loop
reverse its direction
that will be an inverse to it right
hidden bough
yeah
long matrix
Alright, so maybe u can then "identify" all the different classes of loops we have in this space
there's like a "simple" way to describe them
hidden bough
the identity, the ones that go around the origin one way, and the ones that go around the origin the other way?
long matrix
well what about the loop i drew composed with itself
lets call it x
x^2 is not equal to x
i dont know if u can see that
long matrix
oh, good observation.
hidden bough
so they are all of the form x^n right
hidden bough
yeah
long matrix
Are you convinced there aren't any other loops we can have
hidden bough
yes
long matrix
ok so have u seen group presentations?
I might write this group as <x>
which is isomorphic to Z
long matrix
yh yh
long matrix
And that's it. For the basics of a fundamental group
btw, proving this thing is actually Z 
its not totally trivial id say
hidden bough
really?
long matrix
Hatcher's algebraic topology
yeah i wouldnt say so...
his proof uses the idea of a covering space
and other sht
but anyways
long matrix
Ok, so other simple examples u can consider
R^2 with 2 holes
or n holes
U dont have to come up with names for these groups
i dont think u are able to at this stage (like u havent seen them)
but you can kinda see how things get complicated
hidden bough
or n holes
would this have n generators
long matrix
yeah....
hidden bough
ok
long matrix
so things start getting cursed when u consider R^2
minus the grid of integer coordinates
hidden bough
oh...
long matrix
And very cursed (this had me for a good while)
when u consider R^2 - Q^2
And the maybe surprising thing is that even though we're still subtracting a countable number of points
this fundamental group is way bigger than R^2 - Z^2
in terms of cardinality and whatever
long matrix
Im not sure if generators is the right way to think about it any more
so one issue is
when we talk abotu generators in group theory
We can only consider finite combinations of them
we cannot consider abcdef.... an infinite product of them
This infinite product, however, is possible in R^2 - Q^2
long matrix
So, we can no longer identify holes with generators
no longer properly generating
Its because of how "loop" is defined
that will allow this sorta thing
hidden bough
hmm
long matrix
But anyways, algebraic topology is nice overall
hidden bough
this is cursed š
long matrix
I like to think this is cursed thanks to how R works
which is a symptom of analysis
the continuum
hidden bough
which is a symptom of analysis
ok well i guess i have a grudge against analysis too
any recommendations on where to see more alg top
long matrix
Its a good subject until all the edge cases... is how ive felt
So, Hatcher is the textbook that a lot of ppl go for
its like free
published officially
on his website
long matrix
But the prerequisite is a decent understanding of group theory
not too deep, but certainly whats covered in a 1st yr course
which will include quotient groups, actions
hidden bough
i know those
long matrix
and he will explain group presentations
but its good to know those beforehand
Then apart from that, point set topology
Like tbf, its recommended to have understood a 1st yr analysis course
and metric spaces
before touching that
hidden bough
Like tbf, its recommended to have understood a 1st yr analysis course
oh
should i take analysis first then
long matrix
yh
hidden bough
alright
long matrix
algebraic topology was a 3rd/4th yr course for where i was at
not saying that necessarily means its hard, but somewhat advanced
hidden bough
yeah
long matrix
But at the same time, its a lot of pictures, so imo definitely intuitive
hidden bough
i might be able to take analysis over the summer
so i guess i might be able to start next year
hidden bough
i joke about bashing on analysis, but everyone definitely needs it
yeah
long matrix
that along with algebra
neat lintel
explaining makes you better at things
Richard Feynman
hidden bough
i dont pass up chances <:hehe:1080600298604724354>
probably why youre so good at math
neat lintel
He used to say that
hidden bough
He used to say that
does sound like a feynman thing
long matrix
would this have n generators
i was mulling on this tryna remember. Think its precisely the free group on n generators.
That means you can create any "words" or strings with those n symbols (and their inverses)
so, if n = 3
stuff like ab^-1cbc^-1bcb^-1 will be your elements
šµāš« thats hopefully right
long matrix
šµāš« thats hopefully right
importantly, you need to make sure no 2 strings are equal unless they directly cancel via inverses/identity (consecutive aa^-1, etc)
hidden bough
stuff like ab^-1cbc^-1bcb^-1 will be your elements
oh because its not commutative
right
long matrix
yh weird huh
go around one hole, then another
is different from doing vice versa
to advertise a survey you must get permission from @polar panther first
hidden bough
please don't advertise
jagged dock
Okay, how do I get a permission
long matrix
dm modmail
long matrix
Personally, I would say a barrier to really 'getting' algebra and topology is understanding quotients
burnt ledge
And you might want to also tell people before hand its only for 16-20 year olds š
long matrix
what is it?
So, an important topological idea
and analytical tbh
is the idea of localness
locality
local
whatever
you only care about the points in some region
(neighbourhood of some point)
jagged dock
Sorry, I though IB is mostly done at high school
hidden bough
(neighbourhood of some point)
ok
long matrix
So, we might look at only the points in a certain radius of the origin. An example.
If u've done limits and continuity in highschool, taht might have been introduced
nvm if not.
hidden bough
yeah
long matrix
the derivative does not depend at all on slopes far away
Ok, so thats localness
Now, manifolds are spaces who are "locally" "euclidean" everywhere
so, euclidean means R^n, lets say
so everywhere locally, your space looks like (is homeomorphic; aka topologically equal) to R^n
for some constant n
Thats all.
split pawn
Mhm manifolds 
hidden bough
oh
long matrix
So a torus is a manifold
hidden bough
ok
long matrix
a 2-manifold (referring to the surface here)
x^2 = y^2 is not a manifold
the intersection at the middle is problematic
it isn't like a line
R^1
hidden bough
ok
long matrix
You can quite easily generalise the notion of paths, loops intuitively
without getting into the technical topological definitions
Within a manifold
And thats it 
long matrix
Oh yeah, i never said what the point of the fundamental group was
hidden bough
youre a very good teacher
long matrix
defining it
motivations for definitions are always good 
Its so mathematicians can quantify differences between various topological spaces
hidden bough
ohh
long matrix
If they have different fundamental groups, they are different
And you can quantify how they're different too
So R^2 with 2 holes in it, is different from R^2 with 3 holes in it
Unfortunately, it doesn't fully do the job
The (surface) of a sphere has trivial fundamental group, but so does R^2
So to deal with this you introduce more notions that tell them apart
homology... 
Without these kind of notions, it's not easy to work with these objects. you can handwavily talk about them
But really, you want to quantify their differences in some way
hidden bough
The (surface) of a sphere has trivial fundamental group, but so does R^2
is there a reason to differentiate them if they work the same?
is it because one is infinite and the other isn't?
long matrix
Well... there's some fundamental ways the sphere is different from R^2
So... topologically
the interior of a square without a border
is the same as R^2
its not about infinite thingies
necessarily
deep mango
neat lintel
Brushing aside the technical topological space requirements you couldāve imposed, Iām a little curious about why you didnāt mention that manifolds may need multiple charts to represent the whole thing. Or how the chartsā maps should compose themselves along the intersection. The torus is parallelizable
long matrix
(or the interior of a circle without a border)
deep mango
necessarily
"compact" versus "noncompact" is the right thing to say
long matrix
ryc, they'er a 14 yo hser
deep mango
R^2 can be infinite. The sphere cant
long matrix
I dont think I can explain compactness
deep mango
There's no way to pull the sphere so that it's infinite
deep mango
I dont think I can explain compactness
Im doing it rn
long matrix
not very well at least --- thats also to do with my understanding of it tbh
raven plaza
You can't put a sphere into R^2 without it overlapping itself
hidden bough
There's no way to pull the sphere so that it's infinite
so compactness is the "infiniteness"?
long matrix
yh...
but the thing is. Compactness tells you the difference between the open disk (interior of a circle without the border) and the closed disk (with the border)
raven plaza
You can't put a sphere into R^2 without it overlapping itself
Though you can obviously put the plane in itself or the sphere in itself without overlap
deep mango
so compactness is the "infiniteness"?
Compactness means no matter how I try to distort you you'll still be nice and bounded
hidden bough
Compactness means no matter how I try to distort you you'll still be nice and bo...
whereas that isn't true with R^2 say
ok
hidden bough
but the thing is. Compactness tells you the difference between the open disk (in...
ok makes sense
long matrix
the open disk is 'infinite' in a sense (precisely in the sense of non-compactness) 
deep mango
Yeah. Like, if you distort the interval (0, 1) by applying the function 1/x to it
You get (1, infinity)
But you cant do that with [0, 1]
hidden bough
wait why
long matrix
If you add 'infinity' to the real number line (and join the ends into an 'infinite circle') this thing is now compact.
deep mango
Without ripping it into pieces
long matrix
ok ill let ryc lol
deep mango
isn't one a subset of the other
Yeah, but if you go to infinity as x goes to 0, where would you send 0?
hidden bough
oh ig its just not well defined because of division by 0?
deep mango
Ya
long matrix
yeah so small in a different sense. not the literal sense. unfortunately
compact spaces are smaller in the sense of being compact
deep mango
Anyway just one of many ways the sphere and R^2 are different
neat lintel
hidden bough
If you add 'infinity' to the real number line (and join the ends into an 'infini...
weird
long matrix
Anyway just one of many ways the sphere and R^2 are different
surely theres a more intuitive difference
hidden bough
but it's not anumber
deep mango
surely theres a more intuitive difference
To me "bounded versus unbounded" is the clearest and then you just need to fix what you mean by bounded
long matrix
but it's not anumber
yeah, but in topology we dont care about the arithmetic of the number line
theyre all just points
hidden bough
yeah, but in topology we dont care about the arithmetic of the number line
but if we add "infinity" don't we just get the same thing as if it was unbounded?
long matrix
(well until u get to topological groups but anyways...)
raven plaza
why would [0,1] get something smaller
When you're allowed to bend and deform shapes like this, "large" and "small" in the usual sense is gone
deep mango
I also like the cut point approach. If you remove a point from the plane, loops around that point cant be contracted to nothing
But if you remove a point from the sphere, just pull a loop around that point over to the other side of the sphere
To contract it
long matrix
I also like the cut point approach. If you remove a point from the plane, loops...
this would be my preference
yes, just remembered
hidden bough
But if you remove a point from the sphere, just pull a loop around that point ov...
oh ok
deep mango
That goes back to fundamental groups
hidden bough
that makes sense
deep mango
Even though pi1(R^2) = pi1(S^2)
pi1(R^2 minus a point) is different from pi1(S^2 minus a point)
long matrix
I havent said, but basepoint actually doesnt matter for fundamental groups of path-connected spaces. You can "shift it anywhere" and get the same group.
Thats something u prove after defining it ofc.
neat lintel
In my head I immediately associated the punctured plane, via homeomorphism, to S^1xR and cheated
hidden bough
I havent said, but basepoint actually doesnt matter for fundamental groups of pa...
i mean it makes sense intuitively
neat lintel
I did enjoy Shuriās explanation tho
hidden bough
i assume proving it is kinda hard though
long matrix
i think thats easier
than showing fg of that thing is Z
r2-0
actually, the proof is intuitive. I would say. You can handwave it.
hidden bough
really?
long matrix
Yh.
might interest u to have a go...
like just convince yourself
that you can shift the basepoint and get the same fg
thats a proof.
The exact same proof idea is used formally.
===
but oof tryna eli5 compactness. even tao's explanation like idk how to put it
it feels like u rlly gotta at least have some analysis down to get it
It feels unavoidable
deep mango
Only cause you feel a need to actually understand the real definition...
long matrix
but if we add "infinity" don't we just get the same thing as if it was unbounded...
This kinda issue
hidden bough
that you can shift the basepoint and get the same fg
well i mean, if you place the basepoint anywhere, you still only have the identity loop, clockwise loop, and counterclockwise loop i guess
long matrix
My approach to explaining these things would be the formal defn unfortunately. tbh
i cant see another way
like u said yh
neat lintel
Tbh Shuri you had me thinking about Van-Kampenās theorem for a hot minute. Then I remembered I need to prove what the fundamental group of S^1 is and thought back to Hatcherās neat maps
long matrix
well i mean, if you place the basepoint anywhere, you still only have the identi...
you need to convince yourself theres a bijection between the loops
and they do exactly the same thing
group isomorphism
deep mango
Shuri approach: "uhhhhh every open cover has unmm... a... a finite subcover? the fuck is that?"
Ryc approach: "compact means its nice and tidy"
long matrix
if i give you one loop with basepoint P, can you tell me which loop it corresponds to with basepoint Q, and how exactly they correspond.
hidden bough
if i give you one loop with basepoint P, can you tell me which loop it correspon...
hm
ok
long matrix
like how do combining them kinda do the same thing
long matrix
Shuri approach: "uhhhhh every open cover has unmm... a... a finite subcover? th...
yes ryc, but nice and tidy means nothing to me !!!
analysts are cranks i tell ya
deep mango
yes ryc, but nice and tidy means nothing to me !!! <:opencry:586078614865444865>
Which is wild to me. I know all the properties I want and compactness is the best way to get them
long matrix
and u can do that to show isomorphisms, but idk if uve seen that
hidden bough
ive seen isomorphisms
long matrix
Which is wild to me. I know all the properties I want and compactness is the be...
im just not on this level of thinking for analysis 
one day, one day
ive slowly gotten compactness more and more over the last yr
me in undergrad: 
deep mango
I got lucky when I was visiting unis in HS i heard two dudes talk about compactness in the hallway and i heard the open subcover defn
long matrix
I can understand the intuitions u give, but I cannot explain it to other ppl
deep mango
And then when i learned it 2 years later i was like "oh duh..." even though i didn't understand shit from the conversation i heard
long matrix
deep mango
Math is crazy like that
long matrix
yeah, i realy think early exposure is important
A lot of stuff has taken years to fully understand for me
I might have not gotten it in UG, but I did later on
hidden bough
uh if phi(x^a)=phi(x^b) then phi(e)=phi(x^{a-b}) which feels like it has to mean x^(b-a)=e and so b-a=0 and b=a because order of x is infinite but i don't know how to show that the only element in the kernel would be the identity
deep mango
Huh I wonder why that is
Maybe because you have someone you treat like a personal assistant always answering your questions for you
long matrix
uh if phi(x^a)=phi(x^b) then phi(e)=phi(x^{a-b}) which feels like it has to mean...
I wouldnt necessarily go to formal algebra
mmmm im tryna think how to say what sort of proof
the usual one is
its totally geometric.
deep mango
It's visual
hidden bough
its totally geometric.
really?
long matrix
Its like
theres an 'easy' way to show
these 2 groups must be isomorphic
totally visual
easy as in convincing
deep mango
It can be written down mathematically but that would never be a useful way to see it
long matrix
You have to kindof relate these 2 loops directly
and how this 'relation' preserves loop composition
hidden bough
uh if you shift the basepoint it still has the same number of loops around 0 and is still in the same direction
long matrix
Like once I've told you how to get from (any loop in picture 1) to (any loop in picture 2)
(lets call this f)
you can do f(loop1) . f(loop2)
deep mango
Wg
long matrix
or do f(loop1 . loop2)
and its obviously the same
But yeah this might be a bit of a challenge. Its hard to explain what kind of visual proof im after. Everything handwavy
long matrix
uh if you shift the basepoint it still has the same number of loops around 0 and...
yeah, this is true, and the 'visual' proof im after justifies this
long matrix
You directly relate the loop based at P to the loop based at Q in some way
long matrix
You directly relate the loop based at P to the loop based at Q in some way
And while it is true there's only 1 type of loop (1 generator) in this particular space, this proof needs to generalise to other spaces.
Where there may be multiple.
hidden bough
hm
long matrix
i use 'relate' like its meaning in natural language
yh idk lol
Perhaps what might help is to consider a more general composition
You can compose paths together
as long as the end point of the 1st and the start point of the 2nd match up
neat lintel
Do you want to prove this product for composition of paths provides us with a group structure?
long matrix
wanna prove basepoint does not change group
for path connected
Ok, maybe its clearer if i phrase like this
hidden bough
we want to prove that fundamental groups with different base points are isomorphic
alpine kindle
are you talking about k theory
hidden bough
Ok, maybe its clearer if i phrase like this
?
hidden bough
ohhhh
alpine kindle
are you talking about k theory
nvm
hidden bough
do we put the loop on q
long matrix
yeah but how
and we need to do it in a way that will satisfy the homomorphism property essentially
Ah right - so to emphasize. Path-connectedness is like definitely needed
for this proof
thats a decently sized hint
hidden bough
oh wait
long matrix
Path-connectedness === theres's a path between any 2 points in the space
hidden bough
theres always a path from p to q
long matrix
yes yes.
hidden bough
and theres always a path from q to p
so theres a loop that goes from q to p and then in the shape of the loop and then back to q
long matrix
ok, but unfortunately, picking any path from p to q, then any path from q to p. Will not work.
almost there
But what you've written allows this
And this won't be what we want
long matrix
basically, I can pick ways that gets me different loops in pi_1(X, Q) [notation for fundamental group of X basepoint Q]
crystal wing
I have a question about the equation of the line y = ax + b. I am learning machine learning and sometimes you see the equation like this: y = ax + bx + cx + b. I donāt get what it means. It looks like you might have different coefficients but the x is the same. So you end up adding all the xās up. What does it mean to write the equations like this? It comes up in relations to linear regression.
hidden bough
how do we justify that there is an identity loop that goes from p to q and visa versa
neat lintel
hi
long matrix
quasi jettyBOT
I have a question about the equation of the line y = ax + b. I am learning machi...
To ask for mathematics help on this server, please open your own help channel or help thread. See #āhow-to-get-help for instructions.
long matrix
hidden bough
well shouldn't it not go around the point
long matrix
well shouldn't it not go around the point
yes u dont want that
hidden bough
so it should be identity
neat lintel
prove that for every integer n there exist a integer m such that 2^m - m is divisible by n
long matrix
So say I pick a certain path from P to Q
neat lintel
can you solve this problem
long matrix
We then need to pick a path from Q to P
neat lintel
please
long matrix
how do we ensure our choice of path does not do something bad
hidden bough
how do we ensure our choice of path does not do something bad
hmmmm
long matrix
The 1st pick is arbitrary and that will be fine
need to be careful about the 2nd pick
hidden bough
oh wait
neat lintel
<#418981682117345288>
its of imo qualifiers
hidden bough
what if we make the second pick the first pick but in the other direction
long matrix
its of imo qualifiers
long matrix
what if we make the second pick the first pick but in the other direction
Exactly!
hidden bough
ohhhh
hidden bough
right
long matrix
and thats exactly the homomorphism we will need
hidden bough
ok
neat lintel
The reversed path
long matrix
This is a homomorphism between the fundamental groups
because we take a loop based at P
we apply this function
(spit out do y, do loop, do y-inverse)
And this gives us a loop based at Q
long matrix
And then showing this is a homomorphism, and that this is bijective completes the proof.
hidden bough
that seems hard to justify
hidden bough
actually surjectivity seems easier
long matrix
This can get a tad technical
But I think the basic defns i dished out just about handle it
Without getting right into the formal defns
hidden bough
oh i see actually
long matrix
You need to use the fact that loops are equal if they 'continuously deform' into one another for injectivity
hidden bough
and then if two loops based at p both map to the same loop based at q
then they have to have the same number of loops around the point and the same direction
so they are the same
long matrix
I would honestly say the more 'interesting' part of all this is convincing yourself of the homomorphism property. Well thats the algebra part of the proof
f(a)f(b) = f(ab)
this in a sense is a pure computation, but find it satisfying with images 
So that's all the pieces of the puzzle needed š
neat lintel
Iām kinda glad I kept those notes in that notebook now. I know for a fact everything was done in order. Itās around here somewhereā¦
I will find it later. What Iāve said so far is off of memory
hidden bough
f(a)f(b) = f(ab)
oh wait isn't this just because any loop based at g can be mapped to a loop with the same properties based at p
so then f(a) should have the same factorization (idk the right word, string?) as a, and same with f(b) and b, so then f(a)f(b) and f(ab) should have the same factorization/string but one is based at p and the other at g
long matrix
best to draw it imo
we've explicitly defined the function f
(and lets presuppose bijectivity --- that allows you some assumptions on what you draw)
yh u can talk about it completely algebraically, and you would essentially be justifying why both things have the same string
Are the same element
hidden bough
best to draw it imo
i dont have anything to draw it digitally
long matrix
oh wait isn't this just because any loop based at g can be mapped to a loop with...
but without formalisation this uh
not quite there
hidden bough
but without formalisation this uh
is pretty bad yeah
long matrix
Well whenever u want rlly
hmm im still pondering on compactness
within R^n, any closed (for intents and purposes - means "includes borders", but thats only for R^n) and bounded set is compact
Everything else isnt compact
neat lintel
Heine-Borel goes further and says itās compact iff itās closed and bounded
long matrix
But i fail to see how this can be meaningful generalized to the notion analysts wanna get at
yh, i mean to say iff
compact = finite
non-compact = infinite
in a special sense is what they wanna say
But you have to kind of clue yourself in to why the open disc isnt compact and the closed disc is then.
hidden bough
but how does placing bounds at infniity work
long matrix
Ah ok, so consider one space at a time
What ive just said is only about R^n
theres no point at infinity
Adding a point at infinity to the real line creates a new space
Which is compact
long matrix
but thats another thing entirely
neat lintel
What do you mean by an open disk is not compact
long matrix
in R^2
hidden bough
What do you mean by an open disk is not compact
its not bounded is it
long matrix
{p : |p| < 1} not compact
neat lintel
What do you mean by the open disk IS compact
long matrix
i never meant to say that
neat lintel
Ah ok
hidden bough
What do you mean by the open disk IS compact
it seems like it isn't?
long matrix
corrected
neat lintel
We just said Heine-Borelās theorem
long matrix
i mistyped above
hidden bough
oh
neat lintel
Thatās ok
long matrix
open disc isnt
closed disc is
hidden bough
yeah
long matrix
(0, 1) isnt
[0, 1] is
so yeah i think functions can clue you in
and is probably one of the first early encounters into this notion
long matrix
cya
hidden bough
bye
long matrix
A = (0, 1)
B = [0, 1]
Consider general functions
f : A -> R
g : B -> R
for example f(x) = x^2 or g(x) = x^2 on those intervals work
Now, what makes B compact, but A not so
Is that all continuous functions on B are bounded
I can cap the absolute value of g(x), but I cannot for f(x) in general
===
You should be able to name a function f that is unbounded
hidden bough
1/x?
hidden bough
I can cap the absolute value of g(x), but I cannot for f(x) in general
but isn't the cap infinity for some functions?
hidden bough
oh ok
long matrix
This is proven in analysis 1
This definition kindof generalises except you then need to consider what continuity means for f : X -> R for sets X that arent just intervals
But yeah, so this idea generalises
You can have unbounded continuous functions on path-connected subsets of R^n iff the subset is not compact.
I believe. Hope I remembered right
And that kindof clues you in to the notion of finite vs infinite
hidden bough
You can have unbounded continuous functions on path-connected subsets of R^n iff...
yeah that makes sense
if the function was unbounded the subset would have to be as well
and if the subset was bounded the function would have to be as well
long matrix
So the (surface of) the sphere is compact
you can consider continuous maps from the sphere to R
as like uh
deforming the sphere in weird but continuous ways (allows intersecting itself) and then project onto the real line
That is what someone said above, just remembered
hidden bough
you can consider continuous maps from the sphere to R
the sphere to R? isn't its surface 2 dimensional
long matrix
You can stretch or whatever with the sphere. Including making sharp bends. And crossing itself. But gotta be continuous deformation and squish into subset of R
hidden bough
oh ok
long matrix
So to be clear when mathematicians say sphere, they mean its surface. 2d
yes ur right
So u gotta squash into 1d space
and its impossible to squash this to make its image unbounded
squash or stretch
You can maybe think - what if I try to stretch it 'infinitely', but that will fail as a function
you have to still choose where the endpoints go
For every point on the sphere, you have to choose where on the real line it will map to
long matrix
You can maybe think - what if I try to stretch it 'infinitely', but that will fa...
And this will kindof require you to map some point to infinity. Essentially.
Think of a process from left to right
The problem is deciding where that end point will go to
youre allowed weird deformations of the sphere including it crossing itself
but no matter what u do, u will be dealing with endpoints
now, this won't be an issue with the open disc
{p : |p| < 1}
you can certainly continuously map this onto R with unbounded image
(you can even make this continuous map surjective)
long matrix
uh yh, so these things are meant to be spheres - like the 2d surface of a sphere (so lives in 3d space)
hidden bough
(you can even make this continuous map surjective)
is it because of uncountably infinite cardinality or...?
long matrix
is it because of uncountably infinite cardinality or...?
its to do with compactness
hidden bough
what would the mapping even look like
hidden bough
its to do with compactness <:opencry:586078614865444865>
hmmm
long matrix
You can do exactly this stretch
without worrying about endpoints
because the endpoints aren't here.
long matrix
No boundary
so you can kindof stretch the interior
to infinity
exactly like how the 1/x map works
for (0, 1)
hidden bough
right
long matrix
And yh, that's about as far as I got with the intuition of compactness tbh
So the topological property that's going on here. Is that continuous maps actually preserve compactness.
The image of a compact set will be compact for a continuous map.
So when I map from a compact set to R, its image must be closed and bounded
hidden bough
So the topological property that's going on here. Is that continuous maps actual...
so im(S) is compact iff S is compact
which is what you said earlier right
long matrix
yes I believe, thanks for the iff
usually lazy on stating that
im also a bit unsure on these things
so like to check its actually an iff first
long matrix
so im(S) is compact iff S is compact
no actually, not true
hidden bough
wait what š
long matrix
im(S) is compact if S is compact
S compact => im(S) is compact
but unfortunately, you can for example map your entire non-compact space to a single point
hidden bough
but unfortunately, you can for example map your entire non-compact space to a si...
oh
of course
long matrix
The single point is compact
and this map is continuous
Right, so I think im missing a thing or 12
S compact => im S compact
Therefore for maps with codomain R;
S compact => im S closed and bounded
Now what if S is not compact
Right its a slightly different statement
===
I wanted to say if T is not compact, its possible for to have a map where im T not bounded
that... should be true
hidden bough
S not compact => im(S) can be not compact?
long matrix
yeah yeah
hidden bough
seems like it should be true
long matrix
so the function is no longer constant
in this consideration
yeah, should be.
id have to go through the formal defns to convince myself tho
long matrix
===
I wanted to say if T is not compact, *its possible* for to have a map where ...
or grab @deep mango
hidden bough
well i'll get there eventually and then i'll be able to do this all formally
just like a year and a half in teh future...
but in the grand scheme of things not that long
deep mango
or grab <@533879085542146059>
Oh
long matrix
Claim that I cba to prove:
Let S subset of some topological space. Let us consider functions from S to Reals (aka real functions on S). Then
[ exist function : im(S) not compact ] <=> S not compact
https://math.stackexchange.com/questions/1471367/continuous-image-of-non-compact-sets ok think this pretty much is it
hidden bough
Claim that I cba to prove:
> Let S subset of some topological space. Let us con...
cba?
hidden bough
oh
long matrix
The => is easy, its the contrapositive of
S compact => im S compact
hidden bough
The => is easy, its the contrapositive of
S compact => im S compact
right
long matrix
The <= will require you to construct that function
and I don't know of a way without going into the formal definition of compactness.
hidden bough
is there some function that would always work
burnt ledge
The <= will require you to construct that function
or just prove it has to exist 
hidden bough
or family of functions
long matrix
well even with the topological spaces ive described to you
which are all manifolds i think
thats very very general to me.
to come up with a kind of function -> R for all of them that works
I don't feel like we have the tools to do it with this level of explanation
hidden bough
I don't feel like we have the tools to do it with this level of explanation <:Ke...
i mean there's gonna be a limit
since you aren't teaching formally
the fact that you got this far with pretty much jsut intuition is really impressive actually
long matrix
if u learn analysis, terrence tao has written some good things
in fact, he probs has a textbook, ive forgotten
and its probably good
hidden bough
if u learn analysis, terrence tao has written some good things
ok
long matrix
he does motivate and intuit well
long matrix
he wrote a pdf on compactness, but unfortunately u probs need analysis to follow it
its good intuition once uve done analysis though
hidden bough
i guess i'll save that for when i finish analysis
long matrix
and yeah, compactness is very important in analysis/top
bane of many, including me
its one of those definitions that just wont make sense right off the bat
for most ppl
tawdry prism
hidden bough
its one of those definitions that just wont make sense right off the bat <:KekHa...
ok
fair enough ig
long matrix
so the other big important thing in undergrad math
is quotients
I always say to ppl learn what a quotient set is first
Thats pure set theory
at an elementary ish level
hidden bough
I always say to ppl learn what a quotient set is first
is that analogous to quotient groups
like equivalence classes
long matrix
yeah those are a type of quotient set
quotient groups are defined on a particular kind of quotient set
hidden bough
quotient groups are defined on a particular kind of quotient set
ok so analysis and some basic set theory
got it
long matrix
itll come over time, but understanding the idea of a quotient is very general
its all the same idea
quotient set, group, ring, space (as in topological)
and more
latent bison
Guys I have a BIG dilemma if someone can please HELP! So I have legit really important specialist maths exam tomorrow however I am currently getting a 15% on every past paper that Iāve done so far (only 3). There are currently 2 options. I could:
- Go to the exam and get a 15%
OR - Skip it and get a derived exam score instead which means getting a doctors certificate
Which one should I do?
ocean harbor
2
neat lintel
blep?
teal lion
all right
neat lintel
okay so it looks like kubuntu does use systemd and it also uses sddm
so sudo systemctl disable --now sddm should do the trick
and reenable it with sudo systemctl enable --now sddm
neat lintel
kills gui
teal lion
i'd still check htop to see what else is running, and kill manually if needed
but if possible, use systemd first
burnt ledge
okay
teal lion
since that's what your distro uses
burnt ledge
will regular top work š
neat lintel
yeah
burnt ledge
wjat kinds of processes should i sus as gui
teal lion
stuff like xorg
neat lintel
kubuntu uses kde yeah
neat lintel
kworker might be kde but not sure
teal lion
kworker seems to be just shorthand for kernel worker threads
burnt ledge
yeag but theres like 300 kworker processes and they all have names like "kworker/7:2-mm_percpu_wq"
neat lintel
a
teal lion
yeag but theres like 300 kworker processes and they all have names like "kworker...
that's kernel stuff
neat lintel
yeah dont kill kworker then lol
teal lion
you'll want to backtrace what it's doing, but killing it doesn't do you any good
but that's annoying to diagnose a lot of the time
because it's often hardware specific
it could be a driver issue
if that's your problem
burnt ledge
okay well . systemd disablinh sddm seems to have made a difference
ive not killed anything since
teal lion
oh, good
burnt ledge
But
Haha just as i said that ive noticed my fans are gradually getting louder
very quiet still but gradually stepping up
Also i cant use my computer without a graphical environment lol so even tho this "fixes" that issue it doesnt really help
teal lion
Also i cant use my computer without a graphical environment lol so even tho this...
i know, it just rules out it being a deeper problem
what hardware are you on?
i'd wait to see if the fans are having problems because of some driver misconfig with your hardware
burnt ledge
whats relevent
msi gs66 stealth laptop
i7-10something-H cpu, nvidia rtx 2080 Super Mobile
teal lion
when was this released?
burnt ledge
2019 i think? give or take a year
teal lion
okay
that uses intel drivers
if it's a graphics issue, and it's linux-specific, it's probably a driver problem
and a nvidia graphics card
so nvidia drivers too
neat lintel
Guys I have a BIG dilemma if someone can please HELP! So I have legit really imp...
Can you tell me what a derived exam score is? I have never heard about this from anybody
long matrix
what are the ways in which a driver can damage your card?
neat lintel
I thought I either go and take the exam, or I take a 0% and watch myself immediately flunk
burnt ledge
So should i just update drivers
That has never fixed any issue in my life ngl but cant hurt
long matrix
tldr: will using a buggy driver damage gpu
latent bison
Can you tell me what a derived exam score is? I have never heard about this from...
Basically we did an general exam in the middle of the year and if you are sick or whatever cannot attend the final exam, you get a derived exam score
So itās basically like an insurance
On your exams
But you donāt know what you got before the final marks get released altogether
teal lion
https://wiki.archlinux.org/title/Xorg#Driver_installation
this is technically archwiki, but you can adapt the packages to what you're using
neat lintel
I am going to do my best to not be upset about this newly explained option I didnāt know aboutā¦
teal lion
https://wiki.archlinux.org/title/Xorg#Driver_installation
this is technically a...
@burnt ledge
burnt ledge
so ur just getting me to install drivers
okay
ill see what happens (it will not make anything better even a little bit)
teal lion
so ur just getting me to install drivers
make sure they're configured right
give the output of lspci
as it asks in the archwiki
neat lintel
archwiki my beloved
it has so much info that applies to a lot of linux distros
even though its focus is arch
:>
latent bison
I am going to do my best to not be upset about this newly explained option I did...
So what do you think would be the better option in your opinion??
teal lion
like how algebra has so much knowledge that applies to lots of fields
even though its focus is algebraic structures
neat lintel
Have you explained your situation to the professor/teacher?
You need to have that handled, first. See what they tell you
burnt ledge
give the output of lspci
neat lintel
Iām not going to get into my situation. Just know that what Iām beginning to realize is going to require a lot out of me to move past
Luckily, I have some time
burnt ledge
im installing xserver-xorg-video-nvidia-525, which is the latest available update
(since KDE Discover just let me know theres updates available)
echo tundra
okay well . systemd disablinh sddm seems to have made a difference
Huh? You disabled sddm
burnt ledge
yeag to see whatwas going on
ive enabledit again obv so i can use thegraphical interface
echo tundra
I had issues with sddm, Downgrading helped me
echo tundra
yeag to see whatwas going on
Ah, I see
echo tundra
I had issues with sddm, Downgrading helped me
You can use the downgrade tool
burnt ledge
respectfully idont think that you are adequately helping me
you dont even know what my issue is, for one
@teal lion idrk what im meant to be doing
vocal vessel
<@&268886789983436800>
Lol they donāt even have anything in their profile.
Bot fail
long matrix
nice profile
neon tundra
Yes, but it becomes more cumbersome.
long matrix
monke
plain island
i can't but True Mods can
long matrix
fake mod
ocean harbor
burnt ledge
man this crap is so annoying
this iswhy i almost never use linux bc theres just Issues
nd now i just feel stupid
teal lion
<@694267416976293979> idrk what im meant to be doing
sorry, i had to take out the dogs
and make brunch
i'm back now
burnt ledge
ah ive booted out of kubuntu alr, bakc on windows
teal lion
it looks like you have more than one graphics card. for some reason i thought you just had the intel i9 processor and a nvidia graphics card.
there's a separate setup for that
if you want this to be stable on linux, you'll want to use the open-source driver variants
nvidia has a history of being a pain on linux
with their official drivers
you're running on an ubuntu/debiam-based distro
plain island
sounds fun but be careful not to get kicked out
plain island
and not to actually cause damage
teal lion
sounds fun but be careful not to get kicked out
this is actually illegal without permission
teal lion
if it would get him kicked out, it's likely illegal anyway
plain island
lol no
teal lion
really?
plain island
depending on the university they can kick you out for much less
or cause a lot of headache at any rate
abstract echo
Guys, i need a programmable graphing calculator. Do you know which is the best for doing calculations and programming?
storm sage
There are a lot of such calculators
The TI-84 is one example, but Casio calculators or other brands will likely be cheaper
abstract echo
The TI-84 is one example, but Casio calculators or other brands will likely be c...
For the TI-84, exits the Python edition
Maybe the Casio FX-CG50 is a good version
wanton sierra
The friend: 
high mirage
IEEE-754 is driving me mad.
torn willow
zealous garden
this iswhy i almost never use linux bc theres just Issues
yeah, skill issues
burnt ledge
Yeag
zealous garden
I used to daily drive Linux
the used to is because I'm currently suffering from a skill issue
burnt ledge
Holy me, thats exactly my situation too
zealous garden
what's your skill issue
storm sage
this iswhy i almost never use linux bc theres just Issues
Hm, when I use Linux, the number of issues I have is small, but the struggle I have to deal with one issue is large
burnt ledge
just cant stand the freaking noise my fans make
storm sage
(Which is why I just don't deal with them :D)
burnt ledge
Hm, when I use Linux, the number of issues I have is small, but the struggle I h...
Yeag same
zealous garden
lmfao
storm sage
If there's too many issues, backup files, delete OS, reinstall OS, issues gone
burnt ledge
my issues are always tiny, few, impossible to solve, and diffocult to overlook
zealous garden
Currently my issue is quite large
the amount of struggle to fix it is like... Medium-High
jagged forge
shoutout to that time i deleted my system python (do NOT do this)
storm sage
shoutout to that time i deleted my system python (do NOT do this)
Lol I did this once in a VM
For a cybersecurity competition, we had to fix all the issues on this one VM machine
Deleted python and lots of issues were fixed...
echo tundra
Reminds me of the time I removed my kernel
zealous garden
My linux partition will not boot, it appears that my SSD has locked itself into Read Only mode, my /var/ partition is corrupted and because the SSD is in RO, fsck can't fix it.
jagged forge
lmao
zealous garden
So I need to get a replacement SSD, image the old SSD onto it, fix my /etc/fstab so that it will boot properly, and then fix my /var/ partition
storm sage
My linux partition will not boot, it appears that my SSD has locked itself into ...
huh....?
burnt ledge
like one time i spent 3 whole waking days, and came close to bricking my laptop, because the boot selection screen on GRUB was 640x480 instead of 1920x1080
jagged forge
TIL what GRUB is
zealous garden
huh....?
step by step then.
- My linux install is on its own SSD.
- This 2TB SSD has a special partition for the /var/ folder, this is where some logs and settings are kept (for example the package installer keeps the cache and records there)
- this partition has been corrupted (possibly because there were intermittent power losses that I have honestly yet to diagnose, hardware skill issue, happens on windows)
- the SSD is dead (stuck in read only mode, will not accept further writes but I can grab data off of it)
- because the SSD is in read only mode, the built in utilities can't fix the likely small error in the /var/ partition
storm sage
like one time i spent 3 whole waking days, and came close to bricking my laptop,...
real
zealous garden
like one time i spent 3 whole waking days, and came close to bricking my laptop,...
I felt you
I managed to get FreeBSD13 running on my Thinkpad with all the functionality working.
All of it
pulsar pagoda
I managed to get FreeBSD13 running on my Thinkpad with all the functionality wor...
how much of it did u compile?
also
i needed to have linux on my laptop and vivado this week
but the week before i had to switch it to windows for the defense of a project from the major
cause mf force u to use windows
fuck my life
zealous garden
that's why I have a spare HDD I keep lying around that just has windows 10 and some essentials installed
I slap it in my (other) laptop whenever I have to do something, like a certification exam I took last year, that needs Windows
ocean harbor
my arkanoid game in java finally worked after 6 days
jagged forge
epic, that takes me back to my younger days as a programmer
ocean harbor
lmao thanks š
im doing this for my java assignment (never done programming until grade 11)
jagged forge
i think that is pretty good for someone whoās new to programming
ocean harbor
you're right 
jagged forge
return out of the main loop and do game = new Game() or something
solar hawk
epic, that takes me back to my younger days as a programmer
You were once young?
jagged forge
what kind of insult is that supposed to be
bro doesnāt know that N is well ordered
solar hawk
what kind of insult is that supposed to be
Itās not
Young people canāt feed themselves and stuff
solar hawk
bro doesnāt know that N is well ordered
Itās not
I can defend my claim
If I need to
jagged forge
okā¦.
solar hawk
Y no ping
karmic goblet
step by step then.
- My linux install is on its own SSD.
- This 2TB SSD has a sp...
Can you do "mount -o remount,rw /" ? Or "mount -o remount,rw /var" if the other mount point.
zealous garden
Can you do "mount -o remount,rw /" ? Or "mount -o remount,rw /var" if the other ...
/ mounts fine... in ro
/, /home/, and my special locations for some other drives/partitions I keep around will all only mount in read only, /var/ will not mount at all, citing the corrupted partition
Viewed in Windows, the whole SSD is set to read only mode
Drive health is great, no known errors
Search results suggest the part is dead
And all testing I can do seems to concur
karmic goblet
So only option is to dd/ddrescue a copy of the corrupt partition and attempt to fix on a writable device?
zealous garden
I'm upset but also it was a 50 dollar Walmart drive I should have known better, and I'm ashamed of myself
zealous garden
So only option is to dd/ddrescue a copy of the corrupt partition and attempt to ...
Yeah, I'm just gonna get a replacement and dd the whole disk
Then fix on the writable drive
Then I have to fix my fstab because I used the UUIDs
karmic goblet
ZFS on Linux (not sure with FreeBSD) is very good for protecting large amounts of one person's data, mdraid RAID1 for OS.
jagged forge
my company used xfs for an app that wrote 2tb of data per day (after compression). we got spurious data corruption
we switched to zfs and now we donāt get corruption anymore
modest thistle
How do I remove the nickname that I set with commands and make it go back to my display name?
sage prawn
I can't cope with school
unbearable
i dont know if I should focus more on math and stuff related with my major or try in more
than few classes
ocean harbor
How do I remove the nickname that I set with commands and make it go back to my ...
nickname in this server or your main nickname, which is included in your settings?
if it's in server then you need dm modmail and request for a nickname or you could do it by having active role
modest thistle
nickname in this server or your main nickname, which is included in your setting...
nickname in another server using the /nick i want to remove it and make it go back to my normal one
obtuse portal
i failed the 8th grade
solar hawk
sorry to hear that <:sadcat:547124024585486359>
implies there will be a next time in therefore he will fail again so he can do it again for the next time.
solar hawk
LOOOOOOOOOOOOOOOOOOOOL
ocean harbor
Meow
ivory forge
Meow
ocean harbor
ivory forge
ocean harbor
ivory forge
solar hawk
This channel has become chill
bronze pelican
solid yarrow
chill 2 has been announced
gusty socket
no way
neat lintel
chill 2 has been announced
Cap
static loom
visual breach
Strange, I thought chill is not getting high traffic
split pawn
Hi d2
muted parrot
how would answer the interview question: "what engineering resources would you bring with you to a desert island?"
deep mango
Chalkboard, laptop with texlive and vscode, coffee machine, "analysis of linear partial differential operators" by hormander, hagaromo chalk
proven knoll
wa er
burnt ledge
ur surrounded bywater
neat lintel
Chalkboard, laptop with texlive and vscode, coffee machine, "analysis of linear ...
One time a classmate brought me some Hagoromo chalk that they bought themselves so I could present a differential geometry problem to the class and I never felt more complimented in my life 
They knew about the saying
^ i had presented other ones for them, before
exotic sparrow
hi guys
ocean harbor
yo
split pawn
Hi
ocean harbor
Reading an English book in English class and it has a lot swearing words 
The book names called "The break by katherena vermette" on chapter 3
undone yew
very PG though, they didn't go into detail about who else is in the uncle's room š¤
ocean harbor
split pawn
Lol
fallow lily
how do i send images
gusty socket
long matrix
@hidden bough Interesting, what I said yesterday about compactness doesn't work for your general topological space. It does work for some though #point-set-topology message
fiery ivy
Quick question, if a function is Riemann integrable, that means the integral is the Riemann sum with n->infinity, right?
What if the function in question is continuous but has infinite variation on any nontrivial interval
Ping when answer pls
burnt ledge
What if the function in question is continuous but has infinite variation on any...
As in, the function is unbounded on every interval?