help me understand compactness

[civic radish]

I don't understand compactness help me

[raw raptor]

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[dry lantern]

What do you not understand exactly ? The definition ?

[timid anvil]

A subset A in any topological space is compact if every open cover of A has a finite subcover

in R^n a subset is compact if and only if it is closed and bounded

there are a myriad of other descriptions as well, e.g every sequence of elements of A contains a subsequence converging in A

[timid anvil]

be more specific

[civic radish]

I understand the definition but I dont understand where it comes from...

[timid anvil]

in euclidean spaces we have closed and bounded sets

e.g closed intervals

a compact set is supposed to generalise this idea to arbitrary topological space

describe a set that doesn't have any "holes" or missing "limit points" without invoking notions like metric or distance

[patent hollow]

hello there

another quite vague statement is that in compact sets we can do most things in finite time kind of

at the end of the day everything is a big tautology and we gave it a name so as to not have to refer to "insert well known definition" multiple times

If you ask the question "where is this even used so much for it to be named?" It'd be better

[civic radish]

oh yeah

answer that

sorry I forgot about this 💀

[patent hollow]

okay here's an example

for 2 sets A, B and some point x define the following distances

$H(A, B) =\inf_{(p, q)\in A\times B}d(p, q)$

[autumn sierra]

Cofailure

[patent hollow]

$H(x, A):=H({x}, A)$

[autumn sierra]

Cofailure

[patent hollow]

now the question arises

when can we actually find some $x\in A$ and $y\in B$ such that $H(A, B)=d(x, y)?$

[autumn sierra]

Cofailure

[patent hollow]

we can do this if A is compact and B is closed!

[patent hollow]

i this case, we can find some p in A such that d(x, p) = H(x, A) if A is closed

but when it is compact we can use the continuity of the distance function f(p) = d(x, p) to claim that this reaches a minima in A

This is another property, might be one of the most commonly used

continuous functions admit maximums and minimums on compact sets

@civic radish you there?

[civic radish]

yes

wait

interesting

[patent hollow]

say we have a problem :

There is a complete G(V) where the vertex are labelled by x_i

the happiness of some edge {u, v} is the product of the labels of u and v

[civic radish]

vertex?

G(V)?

[patent hollow]

the happiness of the graph is the sum of all the happiness' of the edges of G(V)

[patent hollow]

yeah?

[civic radish]

whats G(V)?

[patent hollow]

graph with vertex set V

do you know this notation?

[civic radish]

nope

is this graph theory?

[patent hollow]

the question arises, does a maximal tuple (x_1,...,x_n) exist?

[civic radish]

what

[patent hollow]

okay don't mind it

[patent hollow]

did you get this?

have to read the "open covers" definition for compactness

[civic radish]

yeah i know

[civic radish]

kinda

[patent hollow]

Try to deduce every definition from every other definition in a metric space setting

like show that X sequentially compact <=> open cover criterion <=> complete and totally bounded

[civic radish]

oh I haven't studied compactness in continuity

[civic radish]

I think I've done most of this

but

isn't the sequentially compact -> open cover compact too hard?

my prof skipped it

[patent hollow]

it isn't hard at all

it's comparatively easier than most of the other results regarding compactness

but that's completely subjective

you shouldn't skip it because it might be hard @civic radish, you need to know that to get the whole picture

try to grok the intuition behind proofs too, taking examples in R2 is a good strat

But that is euclidean so try taking some other obscure metrics

[civic radish]

no he said it requires other concepts which we don't have time to cover

[patent hollow]

asking chatgpt for examples is good as well

[civic radish]

and I have a exam tomorrow so I will see it latee

[patent hollow]

you should self study it

[civic radish]

yeah would after the exam

[patent hollow]

here is what you need anyways

: cauchy sequences

[patent hollow]

what's the exam on

[civic radish]

rudin chapters 1,2,3,4 -{connectedness, series}

[patent hollow]

Send me the paper too I want to attend it like an exam

[civic radish]

sure

[timid anvil]

bad idea, you might not recognise mistakes

[patent hollow]

the way of the chatgpt is to search up it's examples online

[timid anvil]

oh lawd ave mercy :/

[patent hollow]

but at that point sometimes googling directly helps

also https://topology.pi-base.org/properties/P000016 just leaving it here

[patent hollow]

@civic radish when can i get the exam paper

[civic radish]

Oh i forgot 💀

I will swnd before sleeping