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## anonymous one year ago help

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1. anonymous

1 Every complete metric space is a ___________ Baire space Blank space Dense space Cardinal space

2. anonymous

Baire space :)

3. anonymous

@Michele_Laino

4. anonymous

i am here sir

5. Michele_Laino

ok! a complete space is a space in which every Cauchy sequence converges in it

6. anonymous

so, which means that Every complete metric space is a ___________ Baire space

7. anonymous

don't get angry at me but what is a Cauchy sequence ?

8. Michele_Laino

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

now, every converget sequence is also a Cauchy sequence

10. anonymous

ok

11. Michele_Laino

furthermore, if each Cauchy sequence converges to an element x of X, since X is complete

12. Michele_Laino

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

ok. thanks for that . now can we answer the questions i asked together sir?

14. anonymous

i noticed it was a typo

15. Michele_Laino

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

which means that a complete metric space is a dense space

17. Michele_Laino

yes!

18. anonymous

thanks so much sir

19. Michele_Laino

ok! let's go to the next question, please

20. anonymous

2 Let (X,τ) be a topological space. If X is second countable, then X is ____________countable Third Second Fourth First

21. anonymous

i think first

22. Michele_Laino

If a topological space is second countable, then it is also first countable, sincce the second countability implies the first countability

23. Michele_Laino

since*

24. anonymous

ok.

25. anonymous

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

i think B

27. Michele_Laino

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

thanks.

29. anonymous

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

please , i don't know things on that

31. anonymous

i think (b,d,e)

32. Michele_Laino

x=a can not be a limit point

33. anonymous

so, whatt should be the correct one

34. Michele_Laino

yes! I think so, it is {b,d,e}

35. anonymous

please why is it (b,d,e)

36. anonymous

please explain sir

37. Michele_Laino

since each neighborhood around x=b, d, e contains points of A other than b, d, e

38. anonymous

ok sir

39. anonymous

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

here it is Q is dense in R, since we can show that between two real numbers, exists a rational number

41. anonymous

i really need your help in this topology. i wish we can make out study time

42. anonymous

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

x=0, 1 can not be isolated points

44. anonymous

is it 0 and 2

45. anonymous

and explain

46. Michele_Laino

no, x=0, is a limit point of A

47. anonymous

so which?

48. Michele_Laino

[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

hmm. so which are the limit points?

50. Michele_Laino

I think the answer is [2]

51. Michele_Laino

since I can find at least one neighborhood around x=2, such that it contains only x=2

52. anonymous

ok thanks

53. anonymous

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

x=0, and x=1

55. anonymous

sir can you teach me four things here??

56. Michele_Laino

yes!

57. anonymous

teach me the difference between usual real line and the standard real line

58. Michele_Laino

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

the so defined "expanded line" is again a totally ordered set

60. anonymous

ok. please teach me the intersection and union of sets of real line. like

61. anonymous

A=(0,1]⋃2

62. Michele_Laino

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

ok

64. anonymous

do they mean (0,2)?

65. Michele_Laino

no, since the set (0,2) is: |dw:1440443474555:dw|

66. Michele_Laino

x=0, and x=2 are not included

67. Michele_Laino

whereas the set A=(0,1] union [2], is: |dw:1440443551194:dw|

68. Michele_Laino

|dw:1440443643274:dw|

69. anonymous

do what do they want us to do?

70. anonymous

what do they want us to do?

71. Michele_Laino

they are representation of the two sets above

72. anonymous

ok

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