## manjuthottam Group Title could someone help me understand Topology of the Reals?! one year ago one year ago

1. manjuthottam Group Title

attached is the solutions for some question relating to it. the first 5 pages deal with it

2. manjuthottam Group Title

i keep getting int and bd wrong

3. wio Group Title

I wish I could help, dunno enough about topology yet

4. manjuthottam Group Title

ok thank you though :)

5. wio Group Title

I can help reason it out if you tell me definitions

6. wio Group Title

Seems like interior is just stripping away the boundery.

7. manjuthottam Group Title

Defn of int and bd: Let S be a subset of R. a point x in R is an interior point of S if there exists a neighborhood N of x such that N is a subset of S. if for every neighborhood N of x, N (junction) S does not = empty set. and N junction of (R\S) does not equal empty set.

8. manjuthottam Group Title

then x is called a boundary point of S.

9. wio Group Title

You understand what neighborhood is?

10. manjuthottam Group Title

well for first problem a) why is the interior empty set while for b) it is the whole set with open parentheses

11. manjuthottam Group Title

and yes i understand what neighborhood is

12. manjuthottam Group Title

what made 1/n interior pt be special to be empty while the union of two sets interior pt is the complete whole thing

13. manjuthottam Group Title

i would have guessed the ans for b) to be 3 or something

14. wio Group Title

Well, the problem is that you have: $1/1,1/2,1/3,1/4$Which is just rational a subset of rational numbers. The rational numbers are no continuous, so neither would be this subset. You can find an irrational number between any two elements, for example. Thus they are all boundary points.

15. wio Group Title

Despite not understanding neighborhood rigorously, that is my reasoning for b)

16. wio Group Title

I mean for a)

17. wio Group Title

And it applies to c) and d) apparently

18. wio Group Title

@manjuthottam make sense?

19. manjuthottam Group Title

so the answer is focusing on rational numbers?

20. wio Group Title

Okay, @manjuthottam do you know what continuity is?

21. manjuthottam Group Title

is it just that the limit exists?

22. wio Group Title

yeah the limit exists at the function

23. wio Group Title

But actually, let's talk about neighborhoods a bit ok?

24. manjuthottam Group Title

ok so neighborhood is x in R. while there is an epsilon greater than zero where |x-y|< epsilon?

25. wio Group Title

Yeah, notice how that looks kinda like a limit.... Anyway...

26. wio Group Title

Consider 1/2, 1/3, and 1/4 Do they have a neighborhood?

27. manjuthottam Group Title

i'm guessing no?

28. manjuthottam Group Title

at least its not that specific of a number

29. wio Group Title

If you think about it, they're equal to 12/24, 8/24, and 6/24 Since you don't have 10/24 or 4/24 in there, you can't really say they make up a neighborhood.

30. wio Group Title

So can you understand how these three points would be boundary points?

31. manjuthottam Group Title

so since there are values in between like 10/24 = (5/12) that are not part of the set,, there is no neighborhoood? ... because it is not a continuous set?

32. wio Group Title

Well, consider if we were to make a function: $f:\mathbb{R}\mapsto\{0,1\}$We let it be $$0$$ if $$x\in S$$ and $$1$$ if $$x\notin S$$ In this case, a neighborhood is just an interval for which $$f$$ is continuous.

33. wio Group Title

In this way, neighborhoods are analogous to continuity.

34. manjuthottam Group Title

hmm ok

35. wio Group Title

Does my analogy sound consistent to you understanding of what a neighborhood is?

36. manjuthottam Group Title

yes it does

37. wio Group Title

Okay so suppose we did have two points that were directly next to each other... they'd have to be $\frac{1}{n}, \frac{1}{n+1}$

38. manjuthottam Group Title

yes i see

39. wio Group Title

Notice how they are equal to: $\frac{2}{2n}, \frac{2}{2n+2}$And the thing is: $\frac{2}{2n+1}$is between these two points.

40. wio Group Title

And since the numerator is even and denominator is odd, we can't reduce this fraction anymore, so it can't be expressed as $\frac{1}{k},k\in\mathbb{N}$

41. wio Group Title

thus it is not in our set

42. wio Group Title

So no matter how big you make $$n, n+1$$, we can still find a number between them... you'll never have any continuity for our function $$f$$ so you'll never find a neighborhood.

43. manjuthottam Group Title

hmm ok that makes sense!

44. wio Group Title

This is sort of a proof by contradiction as to why there can't be any neighborhoods.

45. manjuthottam Group Title

ok, so then based on the defn of int and bd, we have to first see about continuity and neighborhood

46. manjuthottam Group Title

if it is continuous then we can see that int and bd are not an empty set

47. wio Group Title

The continuity thing was more to help me understand neighborhoods than it was to help you. =)

48. manjuthottam Group Title

=D kk but it helped me learn too! lol!

49. wio Group Title

I think what you need to understand is that rational numbers do NOT make any neighborhoods.

50. wio Group Title

you'll always be able to find some irrational number between two rational numbers.

51. manjuthottam Group Title

ok

52. wio Group Title

Thus when they give you a set involving rational numbers or natural numbers, etc. Remember that it must all be boundaries, no neighborhoods and no interior sets.

53. wio Group Title

If you get something in interval notation, just remember that changing [ ] to ( ) should give you interior set.

54. manjuthottam Group Title

ok so even union of sets

55. wio Group Title

There is a trick question in there where they use $$\cap$$ which is the set intersection... in those cases it has to be points in BOTH sets.

56. wio Group Title

Yeah, it should work for unions of sets well too Just not for intersections of sets.

57. manjuthottam Group Title

so like e)

58. manjuthottam Group Title

automatically the int is empty set

59. wio Group Title

For example: $[1,4]\cap [2,6]$This gives us: $[2,4]$this you can change them... $int([2,4]) = (2,4)\quad bd([2,4]) = \{2,4\}$

60. wio Group Title

So for intersections... just compute the intersection. It shouldn't be too bad.

61. manjuthottam Group Title

oh ok, so in sets that have a intersection we apply that method you mentioned but if the intersection is just one point like in e) then int is empty set because there is no neighborhood

62. wio Group Title

yeah

63. wio Group Title

You'd have to give me some $$\epsilon>0$$ but clearly $$2+\epsilon$$ is NOT in the set $$\{2\}$$

64. manjuthottam Group Title

kk thank you so much for helping me !!!!