- #1
Bachelier
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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
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