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anonymous
 one year ago
please can someone explain what open balls, closed balls and spares are in a metric space
anonymous
 one year ago
please can someone explain what open balls, closed balls and spares are in a metric space

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zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6The best way to think about it, so that it makes sense with the name, is with the standard metric on \(\mathbb{R}^2\). Give me a radius \(\delta\) and some point \(a\) and the open ball about \(a\) is all the points within \(\delta\) of \(a\). So its like you surround \(a\) with an open ball.

ikram002p
 one year ago
Best ResponseYou've already chosen the best response.0They are two important conditions in metric I think we should mention them right ?

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.63, he didnt ask about metrics...

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6The concept stays the same through different metrics but they no longer match the name. The taxi cab metric will gave an open ball that looks like a diamond, and the infinity metric a square.... Also note that the actual ball is not the border, its all the stuff inside(for an open ball). So if you take a basket ball and fill it with air, then the open ball is the air(in R^3 with euclid metric). I hope that makes sense.

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6I do not know what a spare is

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0let E =R endowed with the metric Do defined by Do(x,y)= 1, if x is not y and 0 if x =y for arbitrary x,y element of R . compute the ball B(1;1/2)

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6well that says that everything but 1, has a distance of 1 from 1. And we want to know about the points within 1/2 of 1. So there is only one. what is it?

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6Do(1,1.4)=1 Do(1,1.2)=1 Do(1, 0.8)=1 Do(1,1)=0 Do(1,0.6)=1

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1lol it is a weird metric, everything is at 1 unit away from everything

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6how many points have a distance of less than 1?

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6Right, so here is a metric that gives an open ball that is a singleton. Does this make sense? @GIL.ojei ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0So Sir , what is the question asking us to find and how did u get all those point s and equate them to 1 and how was tour conclusion made??

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6\(B_{Do}(1;1/2)=\{x\in R \mid Do(1,x)<\frac{1}{2}\}=\{1\}\)

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6The question wants the set of all points that are within distance 1/2 of 1. But with this metric, everything, except 1, is distance 1 from 1. So the only point in the set is 1 itself.

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6because the distance from 1 to 1 is 0.

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6if this was the euclidean metric we would have the interval (0.5, 1.5)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1i think "n" points require "n1" dimensions for this metric to be valid/used

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6I don't know what you mean.

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6It passes all the rules of a metric on a set.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1at least in euclidean metric in \(\mathbb{R}^n\)... if we have 3 points, then they can be at 1 unit away from each other only if they are at corners of an equilateral triangle  two dimensions

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1similarly if we have 4 points, we must go to 3space where the points can be at vertices of a tetrahedron or something .. its hard to visualize for more points idk lol

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6A metric is a binary operation on the set. It takes only two elements as an argument .

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6err not a binary operation but from XxX to R.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1XxX to R is a binary operation which takes two operands as input and spits out one real number as output right

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6I think a binary operation on X has to have X itself as the codomain

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6\(\circ : X \times X \rightarrow X\)

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6That's a binary operation... but anyway.

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6But I think I see what you are trying to do and that is think about shapes with this metric. I am not willing to take that jump tonight :)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1Exactly! I am trying to visualize, which is forbidden sometimes in real analysis haha!

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6How would we define a square with a normal metric?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1I see your point, taxicab metric works well i think ?

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6I want to think about a square in this metric with that definition.

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6Well not even that. I am just saying how ever we define a square with the normal distance function on R^2, lets use that definition on this metric and try and think of what a square looks like.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1there are only two possible values for distances here : {0, 1}

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6So a square with side length 1, lets say the unit square and look at the point (0,1/2) normally we would have all the points that are 1 unit away in one direction and we get only one point (1,1/2) But with this metric we get EVERYTHING... lol

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6So most shapes will give everything.

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6I think the only purpose of this metric is to ask this question :) ok 5am good night

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1Haha that is really weird to visualize! xD

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0the metric is just the one that induces the discrete topology, so balls of radius \(r<1\) only contain one point: \(B_{r\,<\,1}(p)=\{p\}\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0https://en.wikipedia.org/wiki/Discrete_space#Definitions this is because all the points are isolated

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0please guys, you have been arguing and i do not understand one bit please, what are the steps in solving the quation i gave and what would be the final answer

ikram002p
 one year ago
Best ResponseYou've already chosen the best response.0i need to see the definition in your book of metric space in your book + which class is this to help you more ^_^

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i am in my finals in national open university. here is a link to the book

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0http://www.nou.edu.ng/uploads/NOUN_OCL/pdf/SST/MTH%20401.pdf

triciaal
 one year ago
Best ResponseYou've already chosen the best response.0sorry can't help on this I don't know

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0let E =R endowed with the metric Do defined by Do(x,y)= 1, if x is not y and 0 if x =y for arbitrary x,y element of R . compute the ball B(1;1/2)

dan815
 one year ago
Best ResponseYou've already chosen the best response.0what is the definition of a ball

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0the open ball of radius \(r\) about \(p\) in a metric space \((X,d)\) is defined as $$B(p;r)=\{x\in X:d(p,x)<r\}$$

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yes that was the definition in my book, they gave just open balls, closed balls and sphare

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0but did not define a ball

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0in this case, it doesn't matter whether they ask for open or closed balls, because there are no points other than \(p\) that are up to or within distance \(1/2\) of \(p\), so the ball in our discrete metric is a singleton: $$B(p;1/2)=\{p\}$$

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0remember the definition of the metric here: $$d(x,y)=\left\{\begin{matrix}0&\text{if }x=y\\1&\text{if }x\ne y \end{matrix}\right.$$

Kainui
 one year ago
Best ResponseYou've already chosen the best response.0@GIL.ojei Explain what you understand.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0for example, suppose our space consisted of the following points \(p,q,r\). we know: $$d(p,p)=0\\d(p,q)=1\\d(p,r)=1$$ so the only thing within a distance of \(1/2\) is \(p\), since \(d(p,q)=d(p,r)=1>1/2\) and \(d(p,p)=0<1/2\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0that's the problem you asked about, @ganeshie8 answered it hours ago

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so, how did he get does points like d(1;0.8)=1 I MEAN THE 0.8

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0OK, WHAT ABOUT THE COMPUTATION OF THIS AND SOLVE COMPLETELY WITH STEPS, PLEASE

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0OK, WHAT ABOUT THE COMPUTATION OF THIS AND SOLVE COMPLETELY WITH STEPS, PLEASE

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0let E =R endowed with the metric Do defined by Do(x,y)= 1, if x is not y and 0 if x =y for arbitrary x,y element of R . compute the ball B(1;5)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0can some one please answer

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6You need to tell us when you don't understand us. We were not arguing we were discussing math. What do you not understand?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0does it mean that hat they told us to do is to find points from 1 to <5?

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6Do you understand that \(B(1, \frac{1}{2})\) is the set of all points that are within one half of 1?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yes that is 1<x<1/2 right?

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6So with the normal metric we would get stuff like 0.6,0.7,0.8,0.9,1.1,1.2,1.3

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6no with the standard metric it would be 0.5<x<1.5

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6Do you see that? \(1\pm0.5\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yes, e  neighborhood of 1

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6So this is with the standard metric, but we are not in that metric. In this metric distance works differently than you are used to. The distance between any two different points is 1

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6So now there are no points within 1/2 of 1 (except 1 itself)

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6because everything has distance 1

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0So now there are no points within 1/2 of 1 (except 1 itself) ,, please give mare example on it

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6the distance between 1 and 3 is 1 the distance between 1 and 7 is 1 the distance between 1 and 900000000000 is 1

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6So when you ask me what are all the points within 1/2 of 1, I tell you there is only 1 and that is 1 itself

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6So now you tell me the answer to this question and I will know you understand \(B(56, 0.7)=?\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.01 following my definition of d(x,y)

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6I am asking for all the points within 0.7 of 56.

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6every open ball contains only its center point in this metric

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6if the radius is less than 1

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6ok what about \(B(1, 3)\)?

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6no, we want to know all the things with less than distance 3 of 1, and everything has distance 1 from 1.

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.61<3, so it contains every point

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6If I tell you everything is 0 or 1, and then I ask you what is < 3. you say everything.

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6Sorry if I sound condescending in the way I explain things, I am not trying to. :)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0no sir, its ok . as far as i understand it. you are great. please continue

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6That is about it. Everything has distance 1 from each other. So there will be only two outcomes for open balls \(B(x, r)=\{x\}\) if \(0<r<1\) \(B(x, r)\) if \(r\ge 1\)

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6So what is \(B(a, 3)\)?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i don't know because the first condition seems not to be the answer because r>1 so it is not a

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0is it 3? please don't be angry

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6lol I would never get angry. Ok so the question is this. What is the set of points that are < 3 distance from a Everything has distance 1 from a, and 1<3. So everything has distance <3. So the answer is ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0B(1,5) will be what?

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6everything has distance 1 from 1, we want all the things with distance less than 5 what is the answer?

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6correct, if you change the radius to something smaller than 1, you will get only the center, which in this case is 1

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so, what do you think are some important important points to note down about the balls?

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6If you understand the concept, the rest will follow.

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6What class is this for?

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6Have you dealt with the \(\epsilon \delta\) definition of a limit?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0finals in my university. i am facing a very big challenge

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i have done limit but i don't know if i did indept

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6This is a hard topic and most people struggle with it.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0let E =R endowed with the metric Do defined by Do(x,y)= 1, if x is not y and 0 if x =y for arbitrary x,y element of R . compute the ball B(1;5)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so, if i was to see that question in exam, what will be my steps to solving it?

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6First what is the ball asking for?

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6all the points that are at least distance ? from point ?

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.6We want all the points that within \(5\) of \(1\) but everything is distance \(0\) or \(1\) from \(1\) So EVERYTHING is distance less than \(5\).

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so which means that there are three points to note, if r<1, the point becomes 1 and if x=r, the point becomes zero but if r>1, then the point becomes everything , right?
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