1. Show that the set R\Q of irrational numbers, with usual topology (relative topology from R), is topologically complete.

2. Define a complete metric for R\{0, 1} with usual (relative) topology.

3. Define a complete metric for the usual (relative) topology on R\Q.

4. (a) If (S, d) is a complete metric space, X is a Gδ subset of S, and for the relative topology on X, Y is a Gδ subset of X , show that Y is a Gδ in S.

(b) Prove the same for a general topological space S.

5. Show that the plane R2 is not a countable union of lines (a line is a set {(x, y): ax + by = c} where a and b are not both 0).