Can Open Subsets of Real Numbers Form Countable Unions of Intervals?

[Bachelier]

Question:
Given that any open subset E of the set of real numbers is a disjoint union of open intervals.
Is E a countable union of disj. opn intervls.

Answer:

Yes it is. to show this we need to find a Bijection from the set of natural numbers to E.

E = disjoint U_(i in N) of (a_j , b_j) with j in N and a_j , b_j in R

consider then g: N to E with f(n) = i

this is surely a bijection. Hence |E| = |N| hence E is countable.

?

thanks

 

[disregardthat]

Hint: The rationals are dense in R.

Question:
Given that any open subset E of the set of real numbers is a disjoint union of open intervals.
Is E a countable union of disj. opn intervls.

Answer:

Yes it is. to show this we need to find a Bijection from the set of natural numbers to E.

E = disjoint U_(i in N) of (a_j , b_j) with j in N and a_j , b_j in R

consider then g: N to E with f(n) = i

this is surely a bijection. Hence |E| = |N| hence E is countable.

?



thanks

Is (0,1) countable?