By Eric Amar, R. Gay, T. V. Nguyen
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Let f : X ~ Y be a function. Then the following four statements are equivalent: (a) f is continuous. (b) Whenever (x(n))~=l is a sequence in X which converges to some point x 0 E X with respect to the metric dx, the sequence (f(x(n)))~=l converges to f(xo) with respect to the metric dy. (c) Whenever V is an open set in Y, the set f- 1(V) := {x E X: f(x) E V} is an open set in X. (d) Whenever F is a closed set in Y, the set X: f(x) E F} is a closed set in X. Proof. 2. f- 1 (F) := { x E 0 422 13.
Then the subspace (Y, diYxY) is also complete. Proof. 7. D In contrast, an incomplete metric space such as (Q, d) may be considered closed in some spaces (for instance, Q is closed in Q) but not in others (for instance, Q is not closed in R). 8. For instance, one possible completion of Q is R. 1. 3. 2. 5. 3. 7. 4. 9. 5. Let (x(n))~=m be a sequence of points in a metric space (X, d), and let L EX. Show that if Lis a limit point of the sequence (x(n))~=m• then L is an adherent point of the set {x 7. 10. ) 418 12. 8. 15. For each natural number n, let e