Solutions to homework problems

1 Homework Assignment # 1 1. Show that the -δ definition of continuity of a map f : X → Y between metric spaces is equivalent to the definition of continuity in terms of open subsets. Proof. Suppose f is -δ-continuous (i.e., continuous in the sense of the -δ definition of continuity) and that U is an open subset of Y . We need to show that f−1(U) is an open subset of X. The definition of an open subset U of a metric space implies that U can be written as a union of open balls Bi, i ∈ I. Then f−1(U) = f−1( ⋃ i∈I Bi) = ⋃ i∈I f(Bi),