/sci/ - Science & Math

[Anonymous No.10621951]

R8 my point-set topology proofs /sci/. It’s final exam season, so rip me to shreds please.

[Anonymous No.10622267]

>>10621951
For claim 1 (maybe depending on your def'n of what it means for a subspace to be compact?) you need to be more clear about whether [math]U[/math] is an open cover of [math]A[/math] in the topology on [math]X[/math], which I will call [math]\mathcal{T}[/math] or in the subspace topology on [math]A[/math], [math]\mathcal{T}_A[/math]. To prove [math] A[/math] compact, it needs to be an open cover in [math]\mathcal{T}_A[/math]. But then you need to explicitly pull it back to an open cover in [math]\mathcal{T}[/math], because the open sets in the subspace topology are not necessarily open in [math]\mathcal{T}[/math] However, any open set in [math]\mathcal{T}_A[/math] is the intersection of [math]A[/math] with some open set in the original topology. For each [math]S \in U[/math], there exists some [math]S' \in \mathcal{T}[/math] such that [math] S = S' \cap A[/math]. Thus, [math]U' = \{S' \in \mathcal{T}: S' \cap A \in U\} \cup \{X \setminus A \}[/math] is an open cover of [math]X[/math]. Then you get a finite subcover [math]V'[/math] of [math]\mathcal{T}[/math]-open sets. Taking out [math]X \setminus A[/math] and intersecting the contents with [math]A[/math], you get a finite subcover of [math] U[/math].


For your proof of claim 2, [math]\{U_y: y \in C\}[/math] should not be some arbitrary open cover, it should be the open cover from you using the Hausdorfness of [math]X[/math] to separate [math]x[/math] and [math] y[/math]. So you should not say "let [math]\{U_y: y \in C\}[/math] be a collection of open sets", you should say something "observe that [math]\{U_y: y \in C\}[/math] forms an open cover for [math]C[/math]". Depending on how much your prof cares, you might note that this is a use of AC.

For the stuff in the box at the bottom: compact only implies closed and bounded in the context of a metric space, so you should not say this for an arbitrary Hausdorff space. Saying that a set is bounded doesn't really make sense if you don't have a metric.

[Anonymous No.10622272]

>>10621951
>>10622267

Also good luck senpai, study hard and I'm sure it will go well!

[Anonymous No.10622305]

>>10621951

what a bunch of useless trash

[Anonymous No.10622331]

>>10622267
>le epin relative topology
fuck off you fucking moronic algebraist it's so fucking obvious
>>10622305
stupid fucking idiot what type of garbage math do you like

[Anonymous No.10622359]

>>10622267
>you need to be more clear about whether U is an open cover of A in the topology on X
This does indeed need to be specified at some point, but it's probably been done in the problem/theorem statement
>>10621951
Lookin mostly good, except as others have pointed out the "let {U_y : in C} be an open cover" shouldn't be "let". Style is intelligible (if cramped from the blackboard) and proofs are mostly efficient. Consider using "thus" "hence" and so on in place of => just because the latter suggests "implies" which sometimes leads to ambiguity

As for the "closed and bounded" thing, no, that works fine in [math]\mathbb{R}[/math], but you can't carry over results from real analysis into a general Hausdorff space. Annoying, I know.

[Anonymous No.10622374]

>>10621951
absolute shit, and the author is probably homosexual

[Anonymous No.10622449]

OP here, I can’t wait to sift through these criticisms when I get home lol
I appreciate it

[Anonymous No.10622630]

>>10621951
i cant understand it but depsite your lack of space I know you can improve the overall readability. proud of you, anon.

[Anonymous No.10622635]

>>10622630
Nigger

[Anonymous No.10622638]

>>10622635
Amoral niggers enabling amoral niggers

[Anonymous No.10622653]

>>10621951
I don't know shit about topology and never (I hope) will, just here to support. Come back to shitpost more with us when you're finished, godspeed

[Anonymous No.10622658]

>>10622331
Financial mathematics.

[Anonymous No.10622725]

>>10622658
not science or math

[Anonymous No.10624330]

>>10622272
>>10622630
>>10622653
Thanks for the encouragement /sci/. I'm studying hard and will do my best. And most certainly be back to constructively shitpost with you all.

[Anonymous No.10624369]

>>10622267
>>10622359
These are very helpful ideas thank you. I will make the necessary adjustments.

I don't think I phrased ' let {U_y : in C} be an open cover ' quite like that, but what would be the more appropriate way phrase this kind of statement in claims 1 and 2?

[Anonymous No.10624376]

>>10621951
>hurr durr /sci/ frens plz check my math that 2+2=4
Kys retard

[Anonymous No.10624381]

>>10622305
you would be shocked at the places where topology has applications. try using this wonderful new thing called a 'google search'

>>10622658
>>10622725
math is math, but unless you're doing hardcore econemetrics, accounting and finance is just glorified arithmetic lol

[Anonymous No.10624389]

>>10624376
brainlet detected. kindly fuck off

>>10622635
>>10622638
see pic related

[Anonymous No.10624714]

>>10621951
I rate 1/10 this is not valid proof and not rigorous. You need to consider the case where there is a cover of A which is not a cover of X. I will also note that uUX\A is a collection of sets united with a collection of points and not expressing what you mean.

[Anonymous No.10624718]

>>10622658
jej, brainlet

[Anonymous No.10624720]

>>10624389
>brainlet detected
this is coming from the guy who wants people correct his illegible shit-tier babby tier proofs. next time write them in latex and put them in the SQT thread you retarded attention seeking faggot

[Anonymous No.10624808]

>>10624720
>t. brainlet

[Anonymous No.10625453]

>>10624369
The key point is that it's not just any open cover, it's one that you specifically constructed to have a particular property. Because if you just take an arbitrary open cover of C, you dont know that it doesn't include x.

So you basically just need to say something like "since X is Hausdorff, for each y in C we may select open sets U_y and V_y such that [etc.]". Then observe that the collection of these U_y is an open cover of C.

[Anonymous No.10625475]

>>10624808
not how the meme works

[Anonymous No.10625589]

>>10621951

2nd semester CS student here

I dont understand any of this and probably never will

[Anonymous No.10625626]

>>10624720
you are not smart

[Anonymous No.10625644]

>>10621951
Some good news and some bad.
Good: I think overall you have a firm understanding of the course material (especially if you did this without notes). Your logic for the proofs is good.
Bad: I agree with the sentiments that you shouldn't use "let {U_y}" as this is not some arbitrary open cover, say something like "Note that {U_y: y in c} forms an open cover of C." And my advice overall is great job, I think you'll do great on this test and these 2 questions are really popular exam questions for topology if you're at a low to mid tier school. But, don't let it get to your head. Don't think you're some kind of math genius, because truth be told this stuff is like baby's first proof. Still, be proud of the progress you've made. Just note that going into Academia won't be this easy...

[Anonymous No.10626250]

>>10625644
Thanks for the advice and complements. Yes this is baby tier stuff but I'm doing it to review. The material we've covered in my undergrad topology class, taught by the head of our math department, has been fairly graduate level stuff at a very graduate level pace.

[Anonymous No.10627657]

bump