The "Dirichlet function" is a famous example. Let f(x)= 0 if x is irrational and f(x)= 1/n where x= m/n reduced to lowest terms.

Q2: Define a metric d on ℝ so that the sequence converges to

1.Whatsequence?

Take T to consist of all open sets of R (under the usual topology) except that "1" is added to any set containing "0" and "0" is added to any set containing "1".Q3: Define a topology τ on ℝ so that the sequence Xn=1/n converges

to 0 , and also converges to 1, but does not converge to any other number. Is

τ metrizable?