Boundary of the interior of the rationals

[Design]

Homework Statement

S = Set of rational numbers
Boundary(interior(S)) = ?

The Attempt at a Solution

I have no Idea how to do this, I don't know what interior of the rational numbers are. Maybe you guys could give an example of like the interior of the natural numbers or the boundary of the natural numbers etc?

thank you

 

[Design]

The question is given like this ∂(S^int)
But maybe I wasn't being clear with the notation so I just wrote it in words up there. Please need help on this!

 

[Office_Shredder]

What's the definition of the interior of a set?

 

[Design]

X a point in the set is in the interior of the set if there exist radius r such that B(r,x) is a subset of S

r = radius, B(r,x) the open ball?

I don't see how this definition can help me with the set of the rational number tho because this definition is so abstract and it looks like we are talking about R^1

 

[Dick]

X a point in the set is in the interior of the set if there exist radius r such that B(r,x) is a subset of S

r = radius, B(r,x) the open ball?

I don't see how this definition can help me with the set of the rational number tho because this definition is so abstract and it looks like we are talking about R^1

We are talking about R^1. And in R^1 the 'open ball' B(r,x) is the open interval (x-r,x+r). Can that be a subset of the rationals?

 

[Design]

We are talking about R^1. And in R^1 the 'open ball' B(r,x) is the open interval (x-r,x+r). Can that be a subset of the rationals?

Umm no that cannot be a subset of the rationals since x-r/x+r can equal a irrational number.
Does this mean that the rational numbers have no interior? So what does that mean?
Also can you give a hint about the boundary part as well :)

 

[Dick]

Umm no that cannot be a subset of the rationals since x-r/x+r can equal a irrational number?

What has (x-r)/(x+r) got to do with it? I'm asking you if every number in the interval (x-r,x+r) (that's the set of all numbers s such that x-r<s<x+r) can be rational? If you don't know just say so. Don't make some sort of goofy guess.

 

[Design]

I didn't mean to mean divide, I meant to say that x-r can equal a irrational number and so can x+r. So no all the numbers in that set can't be rational.

 

[Dick]

I didn't mean to mean divide, I meant to say that x-r can equal a irrational number and so can x+r.

Ok, they can. But that's not the point. If you take B(1/4,1/2) then 1/2+1/4 and 1/2-1/4 are both rational. Does that mean every point in (1/2-1/4,1/2+1/4) is rational?

 

[Design]

No it doesn't, when you don't have a common denominator for the radius and the center adding each other for ball.

For instance B(1/65, 1/2)?

 

[Dick]

No it doesn't, when you don't have a common denominator for the radius and the center adding each other for ball.

For instance B(1/65, 1/2)?

This is going nowhere fast. Could you look up what 'dense' means in topology and how that applies to the rationals and the irrationals in R^1? That might ring a bell about something you have been told that you need to solve this problem.

 

[Design]

We actually never covered anything about dense for toplogy. Just one chapter about interior,boundary and closure and an assignment on it. We are moving to limits for Advanced Calculus now (2nd year course).
I'm not really sure how to apply it to the rationals and the irrationals even after looking at a definition. I don't see it how it is related since we haven't established a subset from the rational numbers.

 

[Inferior89]

Is it possible to chose an r>0 so that every number inside the interval x-r,x+r is rational?

 

[Design]

Are you asking for all x if so, Its not possible.

 

[Inferior89]

For any x.

 

[Design]

Is it not possible to choose an r>0 for any x that will the interval x-r,x+r rational.

 

[Office_Shredder]

Of course it's possible. But that's not relevant. The question is, does the set of rationals have any interior points? If so, then the whole interval (x-r,x+r) must consist only of rational numbers. Not just the endpoints

 

[Inferior89]

Read my question again. I didn't ask if x-r and x+r could be made rational but if it is possible to chose a r > 0 so that the interval [x-r,x+r] only contain numbers that is a subset of the rational numbers.

 

[Design]

Of course it's possible. But that's not relevant. The question is, does the set of rationals have any interior points? If so, then the whole interval (x-r,x+r) must consist only of rational numbers. Not just the endpoints

The rational numbers do have some interior points. So what your saying is the interior of the rational numbers is the rational numbers where (x-r,x+r) are being satisfied?

Read my question again. I didn't ask if x-r and x+r could be made rational but if it is possible to chose a r > 0 so that the interval [x-r,x+r] only contain numbers that is a subset of the rational numbers.

I see what your saying now, your taking the ones that can be made rational and making them into a set?

 

[Office_Shredder]

The rational numbers do have some interior points.

No, they don't

So what your saying is the interior of the rational numbers is the rational numbers where (x-r,x+r) are being satisfied?

What do you mean by (x-r,x+r) is satisfied? (x-r,x+r) is an interval in the real numbers

 

[Design]

No, they don'tWhat do you mean by (x-r,x+r) is satisfied? (x-r,x+r) is an interval in the real numbers

So the interior of the rational is the empty set and the boundary of the empty set is the empty set?

 

[Dick]

We actually never covered anything about dense for toplogy. Just one chapter about interior,boundary and closure and an assignment on it. We are moving to limits for Advanced Calculus now (2nd year course).
I'm not really sure how to apply it to the rationals and the irrationals even after looking at a definition. I don't see it how it is related since we haven't established a subset from the rational numbers.

Ok. Forget the technical definition of dense. Do you know that between every two real numbers you can find an irrational number? And between every two real numbers you can find a rational number?

 

[Design]

Ok. Forget the technical definition of dense. Do you know that between every two real numbers you can find an irrational number? And between every two real numbers you can find a rational number?

Didn't know that before, but I know now that you've told me. Also, can't you find more than one irrational, rational number?

 

[HallsofIvy]

Umm no that cannot be a subset of the rationals since x-r/x+r can equal a irrational number.
Does this mean that the rational numbers have no interior? So what does that mean?
Also can you give a hint about the boundary part as well :)

Yes, you had it back here- the set of all rational numbers does not have an interior. The et of all interior points is an empty set. And what is the boundary of the empty set?

Now any number in a set is either an interior point or a boundary point so every rational number is a boundary point of the set of rational numbers.

But you are not done. Are there any boundary points outside the set? Look at the complement of the rational numbers, the irrational numbers. What is the interior of that set? What are its boundary points? (A set and its complement always have exactly the same boundary points.)

 

[Design]

Yes, you had it back here- the set of all rational numbers does not have an interior. The et of all interior points is an empty set. And what is the boundary of the empty set?

Now any number in a set is either an interior point or a boundary point so every rational number is a boundary point of the set of rational numbers.

But you are not done. Are there any boundary points outside the set? Look at the complement of the rational numbers, the irrational numbers. What is the interior of that set? What are its boundary points? (A set and its complement always have exactly the same boundary points.)

The interior of the irrational numbers is the irrational numbers. The boundary of the irrational numbers is idunno...