For which intervals a b in R is the intersection a b n Q a c

For which intervals [a, b] in ]R is the intersection [a, b] n Q a clopen subset of the metric space Q? (R- real numbers. Q- rational numbers)

Subsets of M that are both closed and open are clopen. See also Exercise 92. It turns out that in R the only clopen sets are and R. In Q, however, things are quite different, sets such as fr E Q-v2

Solution

[a,b]Q is a clopen subset if a,b(RQ), since if a,bQ then [a,b]Q won\'t be open.(same as If aQ or bQ then [a,b]Q is not open.)

if a and b are both rational, then [a,b]Q is not open (in Q)

[a,b]Q isn’t open in Q if even one of a and b is rational.

If one allow intervals [a,b] with a>b, then on must include them as clopen irrespective of whether a or b is rational, since they’re all empty.

intervals with b<a are empty, and hence equal to the interval [2,2]Q