Homework 4 for MATH 104 Solutions Problem 1 Pugh, p.116, #23 Solution. Assume h : Q → N is a homeomorphism. Note that N is homeomorphic to a discrete space: Consider ( N , d ) where d is the discrete metric and let f = id. (1 / n ) is a convergent sequence in Q . Since h is a bijection, all terms of h (1 / n ) are distinct. In particular, the sequence h (1 / n ) is not eventually constant and thus, by 2.12 (c), does not converge in N . Hence h is not continuous, contradiction. ± Problem 2 Pugh, p.116, #18 Solution. (a) Let f : M → N be an isometry. We show that f has the ε-δ-property. If ε > 0, let δ = ε . Then for all p ∈ M , d ( p , q ) < δ implies d ( f ( p ) , f ( q )) = d ( p , q ) < δ = ε. (b) Since any isometry is by deﬁnition a bijection, it su ﬃ ces to show that f-1 : N → M is continuous. Let ε > 0. Again, put δ = ε . Let z ∈ N , and assume y ∈ N is such that d ( z , y ) < δ . Let p , q be such that f ( p ) = z , f ( q ) = y Then d ( f-1 ( z ) , f-1
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