Why Are There No Interior Points in the Set of Real Numbers?

[omri3012]

Hallo,

My teacher wrote that:

"The set has no interior points, and neither does its complement, R\Q" where R refers real

numbers and Q is the rationals numbers.

why can't i find an iterior point?

thanks,

Omri

 

[VeeEight]

Let q be an arbitrary rational number. Does there exist a neighborhood of q that is a subset of Q?

 

[trambolin]

How about the fact that I can squeeze a real number between any two arbitrary points of Q?

 

[HallsofIvy]

How about the fact that I can squeeze a real number between any two arbitrary points of Q?

Irrelevant. Do you mean an irrational number? Now that would be relevant.

 

[HallsofIvy]

Hallo,

My teacher wrote that:

"The set has no interior points, and neither does its complement, R\Q" where R refers real

numbers and Q is the rationals numbers.

why can't i find an iterior point?

thanks,

Omri

So "the set" is Q. For p to be an interior point of Q, there must exist an interval around p, [math](p-\delta, p+\delta)[/quote] consisting entirely of rational numbers. For p to be an interior point of R\Q, the set of irrational numbers, there must exist an interval (p- \delta, p+ \delta)] consisting entirely of irrational numbers. There is NO interval of real numbers consisting entirely of rational number or entirely of irrational numbers.

 

[trambolin]

That was what I said anyway, but of course a real is not necessarily rational part got lost along the way... Sorry for that.

 

[omri3012]

thank you for your comments,

I'm sorry but the statement (as i guess you already assume) was:

"The set Q has no interior points, and neither does its complement, R\Q"

thanks

Omri

 

[HallsofIvy]

Yes, that was essentially what everyone was assuming.