MATH 301 Homework 6 - intersection of countably many open...

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Math 301, Homework 6 1. Find all accumulation points of the following sets in R 2 with the usual metric. (a) 1 n + m ; 1 n + 1 m ± : n;m 2 N o : (b) Q Q (c) D (0 ; 1) r f (0 ; 0) g 2. For a set A metric space ( M;d ) and t > 0 ; let A t = f x 2 M : d ( a;x ) < t for some a 2 A: g (a) Sketch the set A t where A = ² ( x;y ) : x 2 + y 2 = 1 ³ for t = 1 3 ; t = 1 and t = 2 ; where we consider A in R 2 with the usual metric. (b) Show that A t = [ a 2 A D ( a;t ) : (c) Show that A t is open. (Hint: part (2) above.) (d) Show that if b is an accumulation point of A; then b 2 A 1 n for all positive integers n: (e) Show that cl ( A ) = \ 1 n =1 A 1 n : 3. We know that union of any family of open sets is open, while the inter- section of only many open sets is always open. (a) Find an example in R where intersection of countably many open sets is not open. (b) Show that every closed set in a metric space can be expressed as an
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Unformatted text preview: intersection of countably many open sets. (Hint: problem 2. part (e) above.) (c) Show that every open set in a metric space can be expressed as an union of countably many closed sets. (Hint: take complements.) (d) Express the interval (0 ; 1) in R as a union of countably many open sets. 4. Let ( M;d ) be any metric space and let A be a &amp;nite set in M: (a) Show that A is closed and bounded. (b) Without using any theorems, just using the de&amp;nition show that A is sequentially compact. (c) Without using any theorems, just using the de&amp;nition show that A is totally bounded. 1...
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