# e2key - I M413 EXAM II Name t-e~ There should be 5 distinct...

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I M413 EXAM II Name t-e~~ There should be 5 distinct pages with a total of 11 problems. . Let (M, d) be a metric space. 1. (2 pts. ea.) State which of the following sets are always open. ~ (a) M t" tIT (b) the intersection of an arbitrary collection of open sets ~ (c) the union of an arbitrary collection of open sets bfw' (d) 0 2. (3 pts.) Explain what "The rationals numbers are dense in JR" means in terms of the disk D(x,e). ~iv~ ')(. b fR} £ '1 0 I ':j ~ f: (Q cst ~ E V (~/ {. .) ,. 3. (2 pts. ea.) State what int(A) is, in the metric space (M, d) where (a) A= {I}, and (M,d) = (R,d(x,y) = Ix-yl) V\A.]\-\f\) = r/J (b) A = {I}, and (M, d) = (JR, discrete metric) ~ (fr) =- A (c) A = {x E (0, 1)lx is irrational}, and (M,d) = (R,d(x,y) = Ix -yj) \(\.t(h)-=- 9 4. (2 pts. ea.) State which of the following sets are always closed. clo~(d (a) M c.\o')~ cf (b) the intersection of an arbitrary collection of closed sets NoT (c) the union of an arbitrary collection of closed sets doSt-J (d) 0 1

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2 5. (3 pts. ea.) State three conditions each of which by itself guarantees that a set is closed. (a) (OM \\.vV\eA-\ \~ ofAP/'" (b) C6V\~~5 o...~\ ~1.vY'I"Il. ..\~ +IM ~+- s (c) rf,)<,.V'\ '\~ ~ ~e1' \'\,1\ ~.s.ei ) .><.~ ~ X -+~ ,. , t. ..L X I ~ J V'\. -"f;"yv. S(. .:.\:. 6. Let xn be a sequence in a metric space (M, d). (a) (8 pts.) What is the difference between the cluster points of the sequence xn and
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## This note was uploaded on 01/26/2010 for the course MATH-M 413 taught by Professor Michaeljolly during the Spring '08 term at Indiana.

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e2key - I M413 EXAM II Name t-e~ There should be 5 distinct...

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