anonymous
  • anonymous
i have another question in metric topology.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
let E=R be endowed with the euclidean metric \[d {2}(X,Y)= \sum_{k=1}^{2}(Xk -Yk)^{1/2} for all X=(x1,x2),y=(y1,y2)\in R ^{2}.. descibe the set B _{0}(0,0);1). and also (2) discribe the open ball B _{0}( (0,0);1)\]
anonymous
  • anonymous
@oldrin.bataku
anonymous
  • anonymous
@zzr0ck3r

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anonymous
  • anonymous
@zzr0ck3r
zzr0ck3r
  • zzr0ck3r
I cant see the full question
zzr0ck3r
  • zzr0ck3r
describe the ..... It goes off the side of the page.
anonymous
  • anonymous
describe the set \[B _{0}( (0,0);1)
zzr0ck3r
  • zzr0ck3r
\(\{x\in \mathbb{R}^2 \mid \sqrt{x_1^2+x_2^2} <1\ \}\)
zzr0ck3r
  • zzr0ck3r
It is a circle around the origin of radius 1. We include everything in the circle, but not the "border" of the circle
anonymous
  • anonymous
2) describe the open ball \[B _{0}( (0,0);1)\]
anonymous
  • anonymous
The Euclidean metric is the normal metric you are used to. We are looking at the open set $$\{ (x,y) : x^2 + y^2 \le 1\}$$, which is just the set of things within distance \(1\) of our point. This is an open disk of radius \(1\)
anonymous
  • anonymous
Er the inequality should read less than
zzr0ck3r
  • zzr0ck3r
<1 @oldrin.bataku
zzr0ck3r
  • zzr0ck3r
word*
anonymous
  • anonymous
I'm on my phone and OpenStudy lags
anonymous
  • anonymous
Okay, it seems open balls of radius \(r\) centered at \(p\) are notated \(B_o(p;r)\)
anonymous
  • anonymous
so, what is the description?
zzr0ck3r
  • zzr0ck3r
yes, in general open is \(B(a,b)\) and closed is \(B[a,b]\)
zzr0ck3r
  • zzr0ck3r
When we give the answer as a set, that is a description @GIL.ojei this one is very basic. Can you explain exactly what you are having a problem with.
anonymous
  • anonymous
you know my book did not explain that but this is just the question.
anonymous
  • anonymous
so, number the answers for me because it is two questions
anonymous
  • anonymous
I have never seen the closed ball notated as \(B[p;r]\), only as \(\bar B(p;r),\operatorname{cl}B(p;r),[B(p;r)]\), but I imagine it looks like \(B_c(p;r)\) in @GIL.ojei's book
anonymous
  • anonymous
ok . thank you sirs
anonymous
  • anonymous
show that the mapping f;R---.R+ defined by f(x)=e^x is homeomorphism
anonymous
  • anonymous
do you know what a homeomorphism is?
anonymous
  • anonymous
that is bijective funtiom f such that f and f^-1 are both continuous
anonymous
  • anonymous
it's bicontinuous, i.e. it's an invertible map that is continuous in both directions
anonymous
  • anonymous
what definition of continuity are you working with? sequential continuity? the Cauchy definition in terms of \(\epsilon,\delta\)?
anonymous
  • anonymous
its topological space question
anonymous
  • anonymous
yes, but are we dealing with \(\mathbb R,\mathbb R^+\) as metric spaces? there are several equivalent definitions of continuity in this case
anonymous
  • anonymous
any easy one will be ok
anonymous
  • anonymous
you need to tell us which one you are using, it's not a matter of one being easy or not :p
anonymous
  • anonymous
Cauchy definition
zzr0ck3r
  • zzr0ck3r
Which is?
anonymous
  • anonymous
@zzr0ck3r i do not know how to prove it
zzr0ck3r
  • zzr0ck3r
If it is a topological question. then I assume we are working with the definition pre image of open sets is open.
zzr0ck3r
  • zzr0ck3r
For sure it is a bijection. Do you know how to prove that?
anonymous
  • anonymous
no
zzr0ck3r
  • zzr0ck3r
onto: Let \(y\in \mathbb{R}^{+}\), then \(f(\ln(y))= e^{\ln(y)}=y\). Note that since \(y\in \mathbb{R}^{+}\) we have that \(\ln(y) \in \mathbb{R}\). one-to-one. Suppose \(f(x) = f(y) \). Then \(e^x=e^y\) and as a result we have \(\ln(e^x)=\ln(e^y)\implies x=y\). The function is a bijection. Do you follow?
anonymous
  • anonymous
yes
anonymous
  • anonymous
so , that is the prove
zzr0ck3r
  • zzr0ck3r
This proves that it is a bijection. We now need to show that \(\ln(x)\) is continuous on its domain. Also I guess you might want to show that \(e^x\) is continuous on \(\mathbb{R}\).
zzr0ck3r
  • zzr0ck3r
Do you need to show they are continuous?
zzr0ck3r
  • zzr0ck3r
Or just go from the known fact that they both are?
anonymous
  • anonymous
just go from the known fact that they both are
zzr0ck3r
  • zzr0ck3r
Then you are done.
zzr0ck3r
  • zzr0ck3r
You can google how to prove they are continuous, and if I prove it, it will look exactly like any proof you find online.
anonymous
  • anonymous
waw
zzr0ck3r
  • zzr0ck3r
what is waw? I see you make that comment before.
anonymous
  • anonymous
i mean you are a genius
zzr0ck3r
  • zzr0ck3r
not at all...
anonymous
  • anonymous
i wish i am good like you
zzr0ck3r
  • zzr0ck3r
keep at it. Go back and look at the questions I asked 2 years ago. They are the same questions you are asking
zzr0ck3r
  • zzr0ck3r
1 year ago*
anonymous
  • anonymous
please show they are continuous but before then, the last quation for discribing a set , which is open ball and which is set B_{0}
anonymous
  • anonymous
?
zzr0ck3r
  • zzr0ck3r
I am not sure if they want a description like you and I think of when we describe things, or a mathematical description which is a term we use to when referring to a set and a relation on that set. Case 1: visual description: Consider a circle at the origin with radius 1, it is all the points inside the circle. |dw:1438991405420:dw| In this picture, it is all the "white" inside the circle. Case 2: Mathematical description: \(B_0((0,0), 1)=\{(x_1, x_2)\in \mathbb{R}^2 \mid x_1^2+x_2^2<1\}\)
zzr0ck3r
  • zzr0ck3r
Here is a good read on showing things are continuous using \(\epsilon-\delta\) proofs. http://people.bath.ac.uk/sej20/docs/epsilondelta.pdf Let me know if you have any questions.
anonymous
  • anonymous
ok. got it . i have another quation
zzr0ck3r
  • zzr0ck3r
close this and ask a new one.
anonymous
  • anonymous
ok

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