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anonymous

  • one year ago

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  1. anonymous
    • one year ago
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    1 Every complete metric space is a ___________ Baire space Blank space Dense space Cardinal space

  2. anonymous
    • one year ago
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    Baire space :)

  3. anonymous
    • one year ago
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    @Michele_Laino

  4. anonymous
    • one year ago
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    i am here sir

  5. Michele_Laino
    • one year ago
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    ok! a complete space is a space in which every Cauchy sequence converges in it

  6. anonymous
    • one year ago
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    so, which means that Every complete metric space is a ___________ Baire space

  7. anonymous
    • one year ago
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    don't get angry at me but what is a Cauchy sequence ?

  8. Michele_Laino
    • one year ago
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    a sequence is said a Cauchy sequence, if given \epsilon>0, there exists a natual number, say N, such thatfor each n, m two natural numbers, greater or equal to N, the subsequent condition holds: \[\Large d_X\left( {{x_n},{x_m}} \right) < \varepsilon \] where \[\Large {d_X}\] is the distance of your metric space X

  9. Michele_Laino
    • one year ago
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    now, every converget sequence is also a Cauchy sequence

  10. anonymous
    • one year ago
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    ok

  11. Michele_Laino
    • one year ago
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    furthermore, if each Cauchy sequence converges to an element x of X, since X is complete

  12. Michele_Laino
    • one year ago
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    oops..I have made a typo, here is the right statement: "each Cauchy sequence converges to an element x of X, since X is complete"

  13. anonymous
    • one year ago
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    ok. thanks for that . now can we answer the questions i asked together sir?

  14. anonymous
    • one year ago
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    i noticed it was a typo

  15. Michele_Laino
    • one year ago
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    since each convergent sequence is a Cauchy sequence, and each Cauchy sequence converges to an element of the space X it self, then X contains all its limits point, in other word, we have: \[\Large \overline X = X\] so X is a dense space

  16. anonymous
    • one year ago
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    which means that a complete metric space is a dense space

  17. Michele_Laino
    • one year ago
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    yes!

  18. anonymous
    • one year ago
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    thanks so much sir

  19. Michele_Laino
    • one year ago
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    ok! let's go to the next question, please

  20. anonymous
    • one year ago
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    2 Let (X,τ) be a topological space. If X is second countable, then X is ____________countable Third Second Fourth First

  21. anonymous
    • one year ago
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    i think first

  22. Michele_Laino
    • one year ago
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    If a topological space is second countable, then it is also first countable, sincce the second countability implies the first countability

  23. Michele_Laino
    • one year ago
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    since*

  24. anonymous
    • one year ago
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    ok.

  25. anonymous
    • one year ago
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    3 If xϵA¯ , then there exists a sequence (xn) of A such that xn→x is only true if X is a(an) ___________________ Countable Metrizable Hausdorff Separation

  26. anonymous
    • one year ago
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    i think B

  27. Michele_Laino
    • one year ago
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    yes! I think so, since in order to speak about limit, we need of a topology, which can be induced by the metric of the space, so we need of a metrizable space

  28. anonymous
    • one year ago
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    thanks.

  29. anonymous
    • one year ago
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    7 Let X=(a,b,c,d,e)andτ=(X,ϕ,[a],[c,d],[a,c,d],[b,c,d,e]).LetA=[a,c]] , then set A’ of limit points of A is given by A′=(b,c,e) A′=(b,d,e) A′=(b,e) A′=X

  30. anonymous
    • one year ago
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    please , i don't know things on that

  31. anonymous
    • one year ago
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    i think (b,d,e)

  32. Michele_Laino
    • one year ago
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    x=a can not be a limit point

  33. anonymous
    • one year ago
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    so, whatt should be the correct one

  34. Michele_Laino
    • one year ago
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    yes! I think so, it is {b,d,e}

  35. anonymous
    • one year ago
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    please why is it (b,d,e)

  36. anonymous
    • one year ago
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    please explain sir

  37. Michele_Laino
    • one year ago
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    since each neighborhood around x=b, d, e contains points of A other than b, d, e

  38. anonymous
    • one year ago
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    ok sir

  39. anonymous
    • one year ago
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    8 Let R , the real line be endowned with the discrete topology. Which of the following subsets of R is dense in R Q Ritself Qc All singletons

  40. Michele_Laino
    • one year ago
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    here it is Q is dense in R, since we can show that between two real numbers, exists a rational number

  41. anonymous
    • one year ago
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    i really need your help in this topology. i wish we can make out study time

  42. anonymous
    • one year ago
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    9 Let A=(0,1]⋃2 be a subset of R . Then the isolated points of AinR are 0 and1 0 and 2 1 and 2 [2]

  43. Michele_Laino
    • one year ago
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    x=0, 1 can not be isolated points

  44. anonymous
    • one year ago
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    is it 0 and 2

  45. anonymous
    • one year ago
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    and explain

  46. Michele_Laino
    • one year ago
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    no, x=0, is a limit point of A

  47. anonymous
    • one year ago
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    so which?

  48. Michele_Laino
    • one year ago
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    [2] is a closed set in R, nevertheless it is an open subset of A, since it is given by the intersection between A and (-1, 4), so I think [2]

  49. anonymous
    • one year ago
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    hmm. so which are the limit points?

  50. Michele_Laino
    • one year ago
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    I think the answer is [2]

  51. Michele_Laino
    • one year ago
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    since I can find at least one neighborhood around x=2, such that it contains only x=2

  52. anonymous
    • one year ago
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    ok thanks

  53. anonymous
    • one year ago
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    10 For the set A in question above, which of the following are the limit points of A ? 0 and1 0 and 2 1 and 2 2 only

  54. Michele_Laino
    • one year ago
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    x=0, and x=1

  55. anonymous
    • one year ago
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    sir can you teach me four things here??

  56. Michele_Laino
    • one year ago
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    yes!

  57. anonymous
    • one year ago
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    teach me the difference between usual real line and the standard real line

  58. Michele_Laino
    • one year ago
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    In general with the real line we indicate the set of the real number without the points: \[ \Large + \infty ,\quad - \infty \] furthermore, when we add those points to the real line, we get the so called "expanded line"

  59. Michele_Laino
    • one year ago
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    the so defined "expanded line" is again a totally ordered set

  60. anonymous
    • one year ago
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    ok. please teach me the intersection and union of sets of real line. like

  61. anonymous
    • one year ago
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    A=(0,1]⋃2

  62. Michele_Laino
    • one year ago
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    they are defined as usually. Namely, the intersection of two sets, is set of all points which belong to both those sets. Similarly for the union of two sets, which is the set of the points which belong to one set or to the other set or to both In your case, I think better is: A=(0,1]⋃[2]

  63. anonymous
    • one year ago
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    ok

  64. anonymous
    • one year ago
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    do they mean (0,2)?

  65. Michele_Laino
    • one year ago
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    no, since the set (0,2) is: |dw:1440443474555:dw|

  66. Michele_Laino
    • one year ago
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    x=0, and x=2 are not included

  67. Michele_Laino
    • one year ago
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    whereas the set A=(0,1] union [2], is: |dw:1440443551194:dw|

  68. Michele_Laino
    • one year ago
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    |dw:1440443643274:dw|

  69. anonymous
    • one year ago
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    do what do they want us to do?

  70. anonymous
    • one year ago
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    what do they want us to do?

  71. Michele_Laino
    • one year ago
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    they are representation of the two sets above

  72. anonymous
    • one year ago
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    ok

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