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anonymous

  • one year ago

i have another question in metric topology.

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  1. anonymous
    • one year ago
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    let E=R be endowed with the euclidean metric \[d {2}(X,Y)= \sum_{k=1}^{2}(Xk -Yk)^{1/2} for all X=(x1,x2),y=(y1,y2)\in R ^{2}.. descibe the set B _{0}(0,0);1). and also (2) discribe the open ball B _{0}( (0,0);1)\]

  2. anonymous
    • one year ago
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    @oldrin.bataku

  3. anonymous
    • one year ago
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    @zzr0ck3r

  4. anonymous
    • one year ago
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    @zzr0ck3r

  5. zzr0ck3r
    • one year ago
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    I cant see the full question

  6. zzr0ck3r
    • one year ago
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    describe the ..... It goes off the side of the page.

  7. anonymous
    • one year ago
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    describe the set \[B _{0}( (0,0);1)

  8. zzr0ck3r
    • one year ago
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    \(\{x\in \mathbb{R}^2 \mid \sqrt{x_1^2+x_2^2} <1\ \}\)

  9. zzr0ck3r
    • one year ago
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    It is a circle around the origin of radius 1. We include everything in the circle, but not the "border" of the circle

  10. anonymous
    • one year ago
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    2) describe the open ball \[B _{0}( (0,0);1)\]

  11. anonymous
    • one year ago
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    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 \(1\)

  12. anonymous
    • one year ago
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    Er the inequality should read less than

  13. zzr0ck3r
    • one year ago
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    <1 @oldrin.bataku

  14. zzr0ck3r
    • one year ago
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    word*

  15. anonymous
    • one year ago
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    I'm on my phone and OpenStudy lags

  16. anonymous
    • one year ago
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    Okay, it seems open balls of radius \(r\) centered at \(p\) are notated \(B_o(p;r)\)

  17. anonymous
    • one year ago
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    so, what is the description?

  18. zzr0ck3r
    • one year ago
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    yes, in general open is \(B(a,b)\) and closed is \(B[a,b]\)

  19. zzr0ck3r
    • one year ago
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    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.

  20. anonymous
    • one year ago
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    you know my book did not explain that but this is just the question.

  21. anonymous
    • one year ago
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    so, number the answers for me because it is two questions

  22. anonymous
    • one year ago
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    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

  23. anonymous
    • one year ago
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    ok . thank you sirs

  24. anonymous
    • one year ago
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    show that the mapping f;R---.R+ defined by f(x)=e^x is homeomorphism

  25. anonymous
    • one year ago
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    do you know what a homeomorphism is?

  26. anonymous
    • one year ago
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    that is bijective funtiom f such that f and f^-1 are both continuous

  27. anonymous
    • one year ago
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    it's bicontinuous, i.e. it's an invertible map that is continuous in both directions

  28. anonymous
    • one year ago
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    what definition of continuity are you working with? sequential continuity? the Cauchy definition in terms of \(\epsilon,\delta\)?

  29. anonymous
    • one year ago
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    its topological space question

  30. anonymous
    • one year ago
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    yes, but are we dealing with \(\mathbb R,\mathbb R^+\) as metric spaces? there are several equivalent definitions of continuity in this case

  31. anonymous
    • one year ago
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    any easy one will be ok

  32. anonymous
    • one year ago
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    you need to tell us which one you are using, it's not a matter of one being easy or not :p

  33. anonymous
    • one year ago
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    Cauchy definition

  34. zzr0ck3r
    • one year ago
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    Which is?

  35. anonymous
    • one year ago
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    @zzr0ck3r i do not know how to prove it

  36. zzr0ck3r
    • one year ago
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    If it is a topological question. then I assume we are working with the definition pre image of open sets is open.

  37. zzr0ck3r
    • one year ago
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    For sure it is a bijection. Do you know how to prove that?

  38. anonymous
    • one year ago
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    no

  39. zzr0ck3r
    • one year ago
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    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?

  40. anonymous
    • one year ago
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    yes

  41. anonymous
    • one year ago
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    so , that is the prove

  42. zzr0ck3r
    • one year ago
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    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 \(\mathbb{R}\).

  43. zzr0ck3r
    • one year ago
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    Do you need to show they are continuous?

  44. zzr0ck3r
    • one year ago
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    Or just go from the known fact that they both are?

  45. anonymous
    • one year ago
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    just go from the known fact that they both are

  46. zzr0ck3r
    • one year ago
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    Then you are done.

  47. zzr0ck3r
    • one year ago
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    You can google how to prove they are continuous, and if I prove it, it will look exactly like any proof you find online.

  48. anonymous
    • one year ago
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    waw

  49. zzr0ck3r
    • one year ago
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    what is waw? I see you make that comment before.

  50. anonymous
    • one year ago
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    i mean you are a genius

  51. zzr0ck3r
    • one year ago
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    not at all...

  52. anonymous
    • one year ago
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    i wish i am good like you

  53. zzr0ck3r
    • one year ago
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    keep at it. Go back and look at the questions I asked 2 years ago. They are the same questions you are asking

  54. zzr0ck3r
    • one year ago
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    1 year ago*

  55. anonymous
    • one year ago
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    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}

  56. anonymous
    • one year ago
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    ?

  57. zzr0ck3r
    • one year ago
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    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: \(B_0((0,0), 1)=\{(x_1, x_2)\in \mathbb{R}^2 \mid x_1^2+x_2^2<1\}\)

  58. zzr0ck3r
    • one year ago
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    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.

  59. anonymous
    • one year ago
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    ok. got it . i have another quation

  60. zzr0ck3r
    • one year ago
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    close this and ask a new one.

  61. anonymous
    • one year ago
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    ok

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