1:06 AM
looked like it had been raining at the duck pond.

(nvm, solved)

1:37 AM
apparently there were sprinkles around the bay area. none in albany as far as i can tell. my wife is camping in ft bragg, hopefully she is not freezing

Wow, I haven't been to Ft. Bragg in ... years.

i love the north coast, but not a fan of driving
my son has his aps so i stayed home to feed him
not sure he even needs his aps, i am impressed by his discipline

He probably can feed himself better than you, too :P
So I read today that the whole UC system will no longer be considering SATs or ACTs. Of course, APs stay. I get the rationale. I'm just so not a fan of the whole AP industry.

i am with you there

I got an email, with no politeness or even a name, saying basically "Here are my questions" about things in your diff geo text. ... "Regards."
I was too nice. I should have called in @copper.

1:44 AM
i really do not understand people.
people do things electronically that they would never dare to do in person

Is this a consequence of the internet? Certainly we can't blame this on COVID. I have no idea even what the status or age of this person is.

i attribute it to the internet.

My experience over the years at UGA was that students were generally polite, whereas helicopter parents tended to the abusive.
(And that's in person, too.)

wow, parent involvement at the 3rd level?

3rd level = college? YES!

1:47 AM
i had that a few times while teaching. and i didn't teach for very long.

wow. i thought one could invoke some over 18 sort of thing, as in i cannot discuss.

one time someone emailed me asking for solutions to my advisor's textbook. "asking" is even kind of an overstatement.

I was quite abused for correcting mistaken advising when a young woman came to me. Her mother berated me on the phone for half an hour. I had parents come in livid when their children didn't exempt whatever course and expected me to say, "Of course. I realize your child is brilliant and I should do whatever you say."
That's at a whole different level of mathematics/society, @leslie.
Yes, @copper, FERPA is a thing.
But since I wasn't their actual instructor in these instances, I'm not sure it applied to that situation.

incomprehensible (parental interference)

FERPA did come up. even for under-18s who happen to be in college.

1:49 AM
Yes, I invoked that if a parent of one of my own students interfered. But when it came to correcting misadvising, I don't see how it's germane.
Or just saying, I can't exempt person X from this course based on what the evidence is. I don't think FERPA is germane.

yeah, advising seems like a different kettle of fish.

you were helping them and they are mad at you???

Well, it was not perceived as help. The misadvice told a CS major that (at that time) she didn't not need to take linear algebra. Many years later, that would have been correct. It was a generic Honors adviser who screwed up. Mommy was furious when I had to countermand it after the student had dropped the linear algebra course.

ah, i see, she was mad at you the university representative as such.

I was the Honors adviser (who should have been advising earlier, obviously). I think in all my 33 years of advising, I screwed up one time (and I admitted to it in the appeal I helped draft).

1:53 AM
Everyone makes mistakes.

Yes, but often advising mistakes are very costly to students. I don't blame the parents for being frustrated, but insisting for 30 minutes that I had ruined her child's life was unacceptable.
And mommy called during office hours when I had 10 students there.
I kept saying I can't talk now. Should have hung up.
Ah, such memories make me relish retirement. But, overall, I loved advising and teaching.

College is much more complicated here, I think.

OK, time to go cook my last non-liquid food for two days.

Enjoy! Too late, but I think you can spend more $$and then there is only a one day no solids wait. and good luck. Wow, someone just downvoted a (not particularly detailed) answer from April, 2013 ... pretty much when I got here. @copper: Yes, that's correct (without money). Procedure is Tues morning. So I exaggerated by half a day. 1:58 AM :-). i got two downvotes on a closed question today. the answer is fine, i suspect at least one is punitive, and since it is closed, probably both. I remember a vendetta against me some time ago, but I don't remember if anything precipitated it. See y'all later. enjoy your dinner! I'm currently computing the degree of a map f:S^1\to S^1 by z\mapsto z^n for n>0. To do this, I compute the local degree for each x_i\in f^{-1}(1). Since f is a local homeomorphism, \deg f|x_i = \pm 1 for each i. But how can I show \deg f|x_i = 1? take orientation into account, or just write down generators and see what the map does to them 2:16 AM Is there a way of finding generator of H_1(U_i,x_i) ? U_i is a small open nbd of x_i @TedShifrin: I seem to have answered a question that you answered a few hours ago. I did not see your answer until I answered because I had left it open on my computer through dinner. I don't think our answers overlap. On second viewing, maybe they are along the same lines (Mean Value Theorem, but then aren't most estimates of the Remainder?) yeah, and the form of the desired result screams MVT. indeed although not loudly enough for me to heed its call This uses the fact that the MVT assures of a c between a and b. I almost gave up at one point until I remembered that since (x-t)^{p-1} vanishes at the end of the interval 2:39 AM Very interesting! Can you think of any applications? @robjohn Hallo Im really close The i's negate every second odd integer! @TedShifrin haven't yet Seems one of those “who cares” results, @robjohn. yep it is not really surprising either so all in all yawn inducing glad I didn't delay dinner for it When i is taken to the power 4n-1, i get -i, and taken to the power of 4n+1 i get -i 2:43 AM the second one is wrong Huh, but thats like : 5,9,13...etc right? yes i^5\ne-i And i^5 = 2i^2 \cdot i what? Oh i meant +i 2:45 AM yes And then all even powers i get -1? no No? Huh i^4\ne-1 Oh every 2nd even power 2:48 AM which ones? 2,6,etc yes So 4n-2 Vs 4n how do you write them? ah yes Sorry, got interupted But tgat is equiv to 2n and 2n-1 And they are the basis for sin and cos series Gtg, Thanks robjohn! 3:05 AM @Ted is tomorrow the day you have to drink unpleasant amounts of fluids? or was that today? i took bart to walnut creek when i had mine, and the timing would have been fine but for a 20 min delay at rockridge. i then had a mile walk on the other end. when i arrived, the staff just pointed towards the bathrooms without asking. it was on a st partricks day too :-) 3:21 AM @robjohn Tomorrow ;). @copper.hat Didn't you need a chauffeur? At least for pick-up? @TedShifrin my wife collected me afterwards, i did not want to drag her for two visits (kids were 7 & 10 then). i am a total wimp with anesthetic and was completely wiped out afterwards. I remember never remembering a word the doctor says. i started chatting with the theater staff and they quickly turned up the taps to shut me up... I imagine shutting you up is a tough task. Evidently I have a long colon. Immediately after the main event, I had to go over to the hospital for a lower GI. Unanesthetized, it was quite painful. I guess because it was right after the anesthetic from the first procedure, they didn't want to give more. 3:39 AM @TedShifrin I suspect many would concur. @robjohn I am squirming at the thought. I have put off my second one a bit for the memory. This us my fifth, if I count correctly. In Da Carmo (p128), he proves a certain self-adjoint operator S = -(\bar{\del}_xN)^T. If X,Y are local extensions of x,y that are tangent to M and N is a local extension of n normal to M, at the last step he shows -(Y,\bar{\del}_XN)(p) = (-\bar{\del}_xN,y), am I correct he evaluating at p and therefore we are back to the vectors ? If so, why is it capital N on the RHS? Also can a single vector be considered a "vector field"? @TedShifrin more than one every 10 years I assume. 4:00 AM @robjohn is it possible (for me) to search the link text in a chatroom? (as in the urls linked in the chat itself.) I don't think so, but i've never tried. @copper.hat you mean like searching for "ucla.edu" in all urls used here? @robjohn i am curious (provoked is probably a better word) how questions get selected to be closed. it seems that they get nominated in cured and shortly afterwards they are closed. i wanted to search for specific questions to see the timeline. i understand much of the closing, but reasonable questions that do not have (easily found) equivalents/dups do not seem prime targets for closing. @copper.hat try https://chat.stackexchange.com/search?q=4141136&room=36 two from Ted and one from me? @robjohn thanks!! i must have goofed, i tried something similar (i thought) in cured a few moments ago. For CURED: https://chat.stackexchange.com/search?q=4140266&room=2165 4:12 AM yep, i am guessing the search for word matches, not partials found it. thanks! yep, it seems that moments after a close nomination the cured vultures all dive in together. doesn't seem quite right. so one person basically can dictate the closing. time for a drink :-) https://imgur.com/a/qrD4DDD So it seems I need to prove that a ball of positive n-dimensional volume has Hausdorff dimension "n" - any hints, please? @robjohn What have I done this time? Those were the comments where we each mentioned that question using the MVT @epsilon-emperor Didn't we do that just not long ago? @robjohn Nope, that was a cube But. A ball contains a … what? 4:24 AM and I don't remember, but if we did it for a ball, it must have been the box-counting dimension - not the Hausdorff dimension @TedShifrin I'm inclined to say many small cubes a cube contains a ball contains a cube contains a.... its a load of @copper.hat Certainly posting a link in chat might increase the question's visibility (and thus likelihood of closure/reopening, deletion/undeletion, etc.) The same goes for posting on meta. But I don't think the word dictate is accureate here. After all, every close voter has mind of their own - and they can decide whether they agree with the suggestion or not. @MartinSleziak thanks! i'll wait. i am bit irked at the moment :-) just seems a pity to lose content. @robjohn Did we? 4:33 AM Moreover, even without mentioning the close/reopen vote anywhere, the question automatically go into the review queue. (Of course, now that it is so big, many of questions in the clove vote review queue go unnoticed.) "Losing content" is a vague term. Clearly, we do not want to keep everything. A an extreme example, I don't think many people are opposed to removing spam. @MartinSleziak it was really more a (now deleted) comment that got to me. i'll be over it tomorrow :-) i understand and am copacetic with the broad goals Wiktionary: copacetic. At least we are already the lucky ones - we can easily get to the list of our own deleted posts. sorry, usa slang @TedShifrin Could you elaborate on this? @epsilon-emperor if A \subset B then H^d(A) \le H^d(B). 4:45 AM @epsilon-emperor: it might be useful to prove that A and rA have the same dimension where r is any positive real. @robjohn That follows from the scaling property of Hausdorff measures! @copper.hat Hmm, I know this. then you know the ball and cube have the same Hausdorff dimension To show \dim_H F \ge n, I'm using exactly the same argument - just that I need to show a ball's dimension must be "n" @copper.hat How? :'( OH maybe i misunderstood, i thought you knew the dimension of one of them? Any cube can be contained in a sphere of radius s\frac{\sqrt{n}}2 4:49 AM If I consider a cube inside a ball, and a cube outside, I can show that the ball's dimension is the same as that of the cube and I do know that a cube has dimension "n" there but hey, I have a question Here (imgur.com/a/qrD4DDD) the author could have used a cube instead of a ball too, right? yes Now the confusing part is What does the fact that F is open have to do anything here? If I have a non-empty set, I can still find a ball or cube inside it? because an open set contains a ball around all its points 4:50 AM even if I have a closed set, it works?? no @robjohn Agreed @robjohn Why not? A line in \mathbb{R}^3 still has dimension 1 and a line is a closed set Hmm, you're right So how about claiming that: the closures of all open sets also have dimension 'n' That seems to work. yes because they contain open sets 4:52 AM Is it true that the only closed sets that have full Hausdorff dimension in R^n are closures of open sets? how does the book you are reading handle sets like the union of a sphere and a line? no. take any closed set union a small closed ball @robjohn Haven't handled it so far - but what's your point? the H^d of a collection of parts is roughly the \max of the H^d of the prts just trying to figure out how they handle mixed dimensions 4:54 AM @copper.hat Hmm this certainly has full dimension i mean that very informally and H^d \le  dimension of the ambient space, of course. @copper.hat Closed ball union some closed set contains the closed ball, which contains an open ball --- all these have full dimension n if the set contains any open ball at all it has full dimension so a tootsie pop has full dimension (sphere on a stick) @robjohn but it is the closure of some open set right? namely the interior? 4:56 AM why but? @epsilon-emperor so the answer to that question is no @epsilon-emperor no @robjohn Why? Oh the stick is one dimensional a line union a ball is full dimension, but it is not the closure of an open set a tootsie pop contains an open set but the stick does not It does make sense now, thanks! I was imagining a cylindrical stick because that's how we get tootsie pops in real life Hehe 4:58 AM it was an ideal toosie pop infinitely long How was Martin able to pop in yet his icon does not show? If you exit explicitly does the icon disappear? @copper.hat watch my icon... @robjohn thanks! @copper.hat Did you mention something to Martin in another room? Just wondering why he popped in. no, but i had a little diatribe in cured @robjohn Am I unwelcome here? 5:05 AM :-) of course you are welcome here If that's the case, I can simply stay away from this room. Although that would make me a bit said. @MartinSleziak no, just haven't seen you here much it is me that they are trying to get rid of I do try not to be seen. lurking in the transcripts? 5:07 AM I look at the chatrooms more often without visiting them. I only enter when I want to say something. But sometimes I have to enter. For example, to look for somebody who can help on MO with ChatJax. @MartinSleziak nobody here can do that ;-) nobody who says nobody I was glad to be able to figure out what had been plaguing Chrome users hello again, it is me, of course, as you well know. can anyone hint at me the proof technique to go from the fact that if a map \Theta from an affine space to a vector space produces a vector whose span is that of a subspace F, the map \Theta maps an affine subspace to F? i got the part that proves the span is that of F, but now, what do I do 5:13 AM @shintuku i have difficulty (it may be just me) parsing your sentence. uh, no, it is most definitely my fault. give me a second @shintuku a vector spans all real multiples of itself \mathcal{F} is an affine subspace of \mathcal{E}, directed by F which is a vector subspace of E. \Theta: \mathcal{F} \to F. I've shown that, for X \in \mathcal{F}, span[\Theta(X)] = F. How do I show \Theta(\mathcal{F}) = F "directed by"? does that mean that all displacements in \mathcal{E} are vectors in E? not real familiar with affine spaces yep, I mean by that that E is the vector space corresponding to the affine space \mathcal{E} 5:19 AM any vector in E is the difference of two points in \mathcal{E}? for any vector, v, in E we can find two points in \mathcal{E}, p_1,p_2, so that \Theta(p_2)-\Theta(p_1)=v? Difference of points in \mathcal{E} isn't defined yet, but for vector spaces it is. I have, \forall A, B, C \in \mathcal{F}, \vec{AB} = \vec{AC} + \vec{CB} @shintuku I thought that was how displacements were defined. equivalence classes of differences of points. Is \Theta an affine map? If I understand, \Theta maps from an affine space into the vector space hm, we're given B = A + \vec{AB}, but difference hasn't been specified yet and proper algebra hasn't been introduced yet well, suppose we're given difference I guess in that case it is a bit more obvious 5:30 AM B-A=\overrightarrow{AB} alright, but then we still only have a span, right? is a span a displacement, not sure. hm, Ill tinker a bit more with this. the author begins the book with affine geometry without coordinates, so I'm completely lost on proof technique So your \mathcal F is given as P+F where P is some point (and F is a given subspace)? The way I've seen the definition of affine spaces, they are defined as the pair (\mathcal P,V), where \mathcal P is a set of points and V is a vector space. \Theta_B: M \mapsto \vec{BM}, where B, M \in \mathcal{F} and \vec{BM} \in F 5:41 AM Ok, so we want \Theta_B(\mathcal F)=F. I was wondering how \Theta(\mathcal F) makes sense. So if all points \mathcal F are of the form B+\vec v, where \vec v\in F, then you get \Theta_B(B+\vec v)= \vec v. So you get every vector in F as an image of some point from \mathcal F. @robjohn A vector space with no distinguished origin. Subtraction of elements is well-defined but addition is not. @shintuku We're looking at page 10 of Audin's geometry, right? yes! exactly bitcoin price right, but wouldn't that still only give us the span, and not the identity \Theta_B(\mathcal{F}) = F? if, as you say, we get every vector in F as an image of some point in \mathcal{F}, then it means, for any M \in \mathcal{F}, span[\Theta_B(M)] = F then, I'm missing the extra step 5:46 AM I see, the point of this exercises is to show that \Theta_B(\mathcal F) is the same for every point B\in\mathcal F. @shintuku I do not really follow what is the problem here. By definition \Theta_B(\mathcal F)=\{\Theta_B(M); M\in\mathcal F\}. I mean, this is the definition of the image of a set. In any case, in Aubin's book \Theta_A(\mathcal F)=F is part of the definition of an affine subspace. The point of this exercise is to show that we get the same vector subspace as \Theta_B(\mathcal F) for every point B. right, using Audin's definitions, I wrote down this: so we can get any \Theta_B(M) as a linear combination of vectors that come from \Theta_A(\mathcal{F}) which means, any \Theta_B(M) can be written using a basis of F but I feel like I haven't proven yet that the map maps the entire affine subspace \mathcal{F} to F, only for an arbitrarily given M I think I must be missing some linear algebra definition somewhere here You have expressed every vector in \Theta_B(\mathcal F) in the form \vec u+\vec v where \vec v is fixed and \vec u\in\Theta_A(\mathcal F). Moreover, \vec v=\Theta_A(B)\in F. So basically this boils down to verifying that if F is a vector subspace and \vec v\in F is a fixed vector then$$F=\{\vec u+\vec v; \vec u\in F\}.$$This also corresponds nicely to the geometrical intuition behind this. If we fix some point A, we'll get vectors of the form \Theta_A(M)=\overrightarrow{AM}. If we fix some other point B and take vectors with the endpoints in \mathcal F, then the difference is exactly the vector \overrightarrow{AB}. (Or \overrightarrow{BA}, depending on the direction we're looking at.) 6:02 AM don't we need to prove F is a vector subspace, or do we immediately get that from showing it is spanned by \vec u + \vec v? It depends on what you denote by F. I am using \Theta_A(\mathcal F)=F, so I know (directly from Aubin's definition) that F is a vector subspace. right of course Since I see that affine space are being discussed here, I will mention that there is also a room called Linear & Abstract Algebra. Although I have to admit that it is usually quite silent. one last question: we've expressed every vector in \Theta_B(\mathcal{F}) in the form \vec u + \vec v. But why does this imply that the vector subspace F is the set \vec u + \vec v does the fixed vector here play an important role? I'm unable to bridge this gap, going from the \vec u + \vec v representation, which to me seems to only denote a span, to the vector subspace both \vec u and \vec v are vectors from F, of course. but how does the fact that we can express every vector in \Theta_B(\mathcal{F}) as a combination of vectors in F prove to us that, actually, \Theta_B(\mathcal{F}) is F itself 6:18 AM I would suggest to forget about the affine structure for a bit and look only at the problem in the language of vector space. Claim 1. If F is a vector subspace and \vec v\in F is a fixed vector then$$F=\{\vec u+\vec v; \vec u\in F\}.$$Is this claim (about linear subspaces) clear, or should we go through the proof of this? It makes sense: it is a vector \vec v to which you add some vector in F Every vector from the set \{\vec u+\vec v; \vec u\in F\} clearly belongs to F. (A subspace is closed under addition.) We also need to show the opposite inclusion F\subseteq\{\vec u+\vec v; \vec u\in F\}. right, that makes sense I.e. we want to show that if \vec x\in F, it can be expressed as \vec u+\vec v for some \vec u\in F. To do this we can simply take \vec u=\vec x-\vec v. @shintuku So is it now clear that Claim 1 holds? yes 6:24 AM If claim 1 (which is the "vector" part of our situation) is clear, then we can look at the "affine" part in our problem. Claim 2. Let \mathcal F be an affine subspace of \mathcal E and A,B\in\mathcal E. Then$$\Theta_B(\mathcal F)=\{\vec u+\overrightarrow{BA}; u\in\Theta_A(\mathcal F)\}.

ahhhhhhhh

Any vector from $\Theta_B(\mathcal F)$ is of the form $\vec x=\overrightarrow{BM}$.
So we get $\vec x=\overrightarrow{BM}=\overrightarrow{BA}+\overrightarrow{AM}=\overrightarrow{BA}+\Theta_A(M)$.
This shows $\Theta_B(\mathcal F)\subseteq\{\vec u+\overrightarrow{BA}; u\in\Theta_A(\mathcal F)\}.$
We need to check also that $\{\vec u+\overrightarrow{BA}; u\in\Theta_A(\mathcal F)\}\subseteq\Theta_B(\mathcal F)$ to finish the argument in this way.

riiiiiiiiiiight

But the manipulation in which we can get this inclusion should be somewhat similar to the first one.

it finally clicked!
thank you so much!
I've spent like three hours on this problem
honestly, thank you for the time spent spelling it out

6:30 AM
I guess this is much easier if one has possibility to draw pictures. (Some plane in 3D representing $\mathcal F$, then drawing points and vectors between them.)
But that would be difficult to do in Stack Exchange chat.

well, I had the intuition down and images and everything
I just couldn't make the logic work out

That's good.

it is surprising to me that we still cannot easily draw & share little pictures on our computers.
key word being 'easily'

I think that what we did here was basically just writing down a formal proof of the same thing one can imagine in the picture for some specific example (such as a plane).

I'll be going back to Audin now, again thank you so much!

6:33 AM
restores my faith in mse humanity

@copper.hat During the last two academic year I have been using a graphics tablet (Wacom) a lot.
Most of my teaching was done in the way that I was writing by hand and talking to students.
But using something like that here in Stack Exchange chat would not work well. I would have to draw the picture and then paste it here.

@MartinSleziak about 2 decades ago i tried to go paperless, but still struggle. i have a pen, used to have a motion computer for a while, but nothing really replaces my pencil & paper :-(

Sequence of chat messages is completely different from sharing a screen, where the other participant(s) can see the things in the way the are added.

that is true, but even static pics can be of big value

That's right I use that only very seldom to write something that is just for me.

6:35 AM
even with work, slack/teams/whatever, there really is nothing that functions well as a whiteboard

But since I had to teach remotely, I had to use something and this seemed as a reasonable way. (With an advantage that I do not need physical blackboard. Some colleagues actually went to an empty classroom and broadcasted what they were writing an a board.)
I have been using Google Meet during the previous academic year, MS Teams this academic year. (The choice of software was basically decided by the university.)

amazing to me that in 2021 we still struggle with basic drawings :-)
i used meet on friday for the first time
it was a meeting with google no surprise :-)

As far as Whiteboard goes, I have experimented a bit with various things, the one I used the most was MS WhiteBoard.

my work is in eda, mostly unix base folks

But most of the time I simply write the thing locally - some shared Whiteboard where both me and the students can write things is needed only somethimes.
Anyway, it was nice chatting here, I guess it's time for me to get some breakfast.

6:39 AM
where are yo???
im about to hit the sack
ok, bye!

closest i've been to there was wien. over 4 decades ago :-(

@copper.hat That's best if said with a British accent... Been to Wien...

such was my naivete that i did not realise that wien was vienna

6:46 AM
In the US, it's like Ben to Wien
just no magic to it

:-)
good night! time for my ugly sleep

Good night, see you later!

@MartinSleziak Later. Thanks for dropping by!
Though I know you'll see us while we don't see you

7:21 AM
I feel this proof in Falconer is incomplete, they only consider one case - i.e. s > \dim_H F
If s < \dim_H F I do realise that H^s(F) = \infty, but we still don't know anything about H^s(f(F))
Oh in that case it seems sufficient to say that the inequality holds trivially. Got it.
One thing still seems off: they say "implying that \dim_H f(F) \le s/\alpha...". It should be without equality, right?
s > \dim_H F and s < \dim_H F cases are handled, but what if s = \dim_H F?

2 hours later…
9:05 AM
when we have an element say $x \in \mathbb{R}$ and the author writes $x > 0$ what does this mean? the scalar product of x with itself bigger than 0 ? each element in the tupel x is bigger than zero? the distance between the element x and the (0,0,0....) is bigger than 0 (in relation to the defined metric)
@MISC {1871279,
TITLE = {Prove that $\mathbb{R}^n\setminus \{0\}$ is connected for $n &gt; 1$},
AUTHOR = {Aloizio Macedo (https://math.stackexchange.com/users/59234/aloizio-macedo)},
HOWPUBLISHED = {Mathematics Stack Exchange},
NOTE = {URL:https://math.stackexchange.com/q/1871279 (version: 2016-07-26)},
EPRINT = {https://math.stackexchange.com/q/1871279},
URL = {https://math.stackexchange.com/q/1871279}
}
Nevermind i think he means each element of the tupel x is bigger than 0

1 hour later…
10:37 AM
@MadSpaces when you want to refer to a post, do not copy and paste the citation from the Cite link. Instead, copy the Share link and use it in a [link to the post](https://math.stackexchange.com/a/1871279) link to the post.
That is a lot less spammy and just as easy to use.

11:12 AM
@epsilon-emperor there is no need for cases
$x\le s/a$ for all $s>y$ implies $x\le y/a$

@AndrewMicallef: did the suggestion for Chrome on Android work for Firefox on Android?

11:29 AM
ah
i just learned about the extensionality axiom
i guess that explains why I couldn't finish most proofs

@Thorgott Ahh, right
Thank you

11:49 AM
is there a way to search back to my last message i sent?

there's a button if you scroll up

aaha so easy thanks

1 hour later…
12:55 PM
@robjohn gotcha.
Guys, in R^2 with the eucledian metric, the open sets can only be balls, right?

do you call $R^2$ a ball?
do you call halfplanes balls?