Homework 04

# Homework 04 - a b(7 Show that a b is incomplete and that a...

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MATH 74 HOMEWORK 4 1 Due Wednesday February 25th at 3:10pm. Throughout, let X and Y be metric spaces. (1) Let x X . Prove that { x } is a closed set, and that the sets X and are both open and closed in X . (2) Show that the constant functions f : X Y , x X, f ( x ) = y 0 , are continuous. (3) A characteristic function for a set A takes the value 1 for elements of A and 0 elsewhere, i.e. χ A ( x ) = ± 1 x A 0 x / A What do we need to require of A for χ A to be continuous? (4) Show that if f : X Y is continuous then for closed sets A Y , f - 1 A is closed in X . (5) Prove that the composition of continuous functions is continuous. (6) Show that (0 , 1) is homeomorphic to (
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Unformatted text preview: a, b ). (7) Show that ( a, b ] is incomplete and that [ a, b ] is complete. (8) Show that if x is a limit point of the set A then there is a sequence { x n } ∞ n =0 in A which converges to x . (Hint: take ±-balls around x , with radii decreasing to 0.) (9) Show that the closed ball, B ± ( a ) := { x | d ( x, a ) ≤ ± } is a closed set. (10) Find an example where the inﬁnite intersection of open sets is not open and one where the inﬁnite union of closed sets is not closed. 1 Comic from www.xkcd.com 1...
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