## anonymous one year ago here

1. anonymous

In the finite complement topology of R, let the sequence [x_{n} be defined by x_{n} = n, for n\epsilon N. If the limit of the sequence is x, then x must be ∞ 0 A unique constant Arbitrary in R

2. anonymous

@zzr0ck3r

3. anonymous

what is the complement of R?

4. zzr0ck3r

what is the set of all elements that are not in R For the 6th times, please stop using epsilon. IT DOES NOT MEAN IN

5. anonymous

i do not know

6. anonymous

i thought there are two types of numbers

7. anonymous

the imaginary and the real

8. zzr0ck3r

when we talk about the real numbers as the universe, we do not consider them as a subset of the complex numbers. The compliment of the real numbers is the emptyset

9. anonymous

ok. thanks so much

10. anonymous

so will that option b empty since is the complement of R?

11. zzr0ck3r

In the cofinite topology, 1) a sequence that does not take an infinite many of the same value will converge to every element in the space. 2) A sequence that does(but only one number), will converge to that number 3) A sequence that has more than one number for which it obtains infinite times does not converge

12. zzr0ck3r

13. anonymous

Arbitrary in R

14. zzr0ck3r

correct

15. anonymous

i wonder how $$\{ x_n,\varnothing ,R \}$$ is considered cofinite :O

16. anonymous

what does arbitrary really mean. because i see it as finite or infinite

17. zzr0ck3r

um, it means if you close your eye and pick one

18. anonymous

Arbitrary = any

19. zzr0ck3r

@halmos? $$\{x_n, \emptyset, \mathbb{R}\}$$ what is this?

20. anonymous

In the finite complement topology of R, let the sequence [x_{n} be defined by x_{n} = n was trying to interpret in notations

21. zzr0ck3r

Arbitrary size means we are not given anything about the set, so as far as order, a set of arbitrary order is either finite or infinite. But we use the term this other way as well :)

22. zzr0ck3r

he sucks with latex :)

23. zzr0ck3r

he also usees $$\epsilon$$ when he means $$\in$$ so you got to be careful with that as well :)

24. anonymous

oh i see now :)

25. zzr0ck3r

Sorry @GIL.ojei I am just playing with you...

26. zzr0ck3r

but seriously start using $$\in$$ when you mean in. It is coded as $$\in) 27. zzr0ck3r \in 28. zzr0ck3r \(\epsilon$$, in general, means a positive real number

29. anonymous

@GIL.ojei u can use this next time for math notations :) http://prntscr.com/8e7ekc and yes epsilon used to denote very small values

30. anonymous

thank u sir

31. zzr0ck3r

or "arbitrarily small" positive number :)

32. anonymous

ok

33. anonymous

Let (X,τ) be a topological space, and let A be a subset of X. A is dense in X if and only if every non-empty open subset U of X, _________________ A⋂U=0 A⋃U=ϕ A⋂U=ϕ A⋂U=X

34. zzr0ck3r

What is that weird symbol?

35. zzr0ck3r

@GIL.ojei ?

36. anonymous

this is just a definition, and i think the notation should be empty set u can use \varnothing $$\varnothing$$ instead

37. anonymous

empty set

38. zzr0ck3r

then what is the 0?

39. anonymous

i think he copied it from a place which typed it as $$\emptyset$$ first (we see it 0) then as $$\phi$$

40. anonymous

its just zero

41. zzr0ck3r

that makes no sense

42. zzr0ck3r

It should be intuitively clear from the definition of a dense set, that if $$U$$ is open then $$A\cap U\ne \emptyset$$. But I do not see this option, and I dont know what the weird $$\phi$$ looking thing is

43. zzr0ck3r

or \emptyset

44. zzr0ck3r

The union of any non empty set and anything else is non empty

45. zzr0ck3r

these options don't make sense.

46. zzr0ck3r

@GIL.ojei please reread the question and see if it shuold say $$\neq \emptyset$$

47. anonymous

Let (X,τ) be a topological space, and let A be a subset of X. A is dense in X if and only if every non-empty open subset U of X, _________________

48. anonymous

A⋂U=0

49. anonymous

that is option a

50. anonymous

A⋃U=$ϕ$

51. anonymous

that is option b

52. anonymous

A⋂U=$ϕ$

53. anonymous

thats option C

54. anonymous

A⋂U=X

55. anonymous

option D

56. anonymous

@zzr0ck3r

57. anonymous

i think is option c

58. anonymous

or option D

59. zzr0ck3r

option a makes no sense the union of subsets of some general topology are not necessarily equal to a number option b is wrong because U is non empty option c is wrong because of what I wrote about the non empty intersection option d is only true sometimes

60. zzr0ck3r

$$A\cap U \neq \emptyset$$ is always true

61. anonymous

Which of these is not true about T1 - spaces? a)Every singleton set is closed b)Every finite set is closed c)Every Hausdorff space is T1 d)Y1isinR

62. anonymous

this is the last question on topology for today. please, i will tell you what to teach me today . please

63. anonymous

i know option A and B are correct

64. anonymous

but what about C and D? i fink C is also correct

65. zzr0ck3r

what does Y1isinR mean and why do you keep writing things like this?

66. anonymous

that was how i saw it

67. anonymous

$Y_1$ is in R

68. zzr0ck3r

you look at the page and it looks like this ? Y1isinR ?

69. anonymous

i think that was what they wanted to write

70. zzr0ck3r

what is $$Y_1$$?

71. anonymous

do not have idea but is Every Hausdorff space is T1?

72. zzr0ck3r

Tell me the definitions of both

73. anonymous

X is a T1 space if any two distinct points in X are separated.

74. anonymous

i know that Hausdorff space has to do with intersection. but what is really separated?

75. zzr0ck3r

what does separated mean?

76. anonymous

77. zzr0ck3r

So if it is Hausdorff, then for all $$x,y$$ we have open set $$O,U$$ such that $$x\in O, y\in U, O\cap U=\emptyset$$ Does this imply that there is a open set $$A$$ that contains $$x$$ and does not contain $$y$$? Does this imply that there is a open set $$B$$ that contains $$y$$ and does not contain $$x$$? IF the answer is yes, then we must be $$T_1$$.

78. anonymous

waw. so, T1 space is also Hausdorff space

79. zzr0ck3r

we are not showing that T1 is Hausdorff, we are showing that Hausdorff is T1.

80. zzr0ck3r

every horse has 4 legs but not every 4 legged animal is a horse i.e. if a implies b it is not necessarily true that b implies a

81. anonymous

ok

82. anonymous

i want to close the tab and open another. i think there are thinks i want to understand