MTH 421 Midterm: Midterm 2 Spring 2009 Solutions
Document Summary
April 1, 2009: determine whether each of the following statements is true or false. If a statement is false, provide a counterexample. (write out the word true or false completely! No proof or explanation is necessary if you answer true, and you don"t need to prove your proposed counterexample is a counterexample if you answer false. ) (a) let f : e rn rm be a continuous function. If h rm is compact, then f 1(h) is compact. If u rn is an open set, then f (u ) rm is an open set. Let f : r r be de ned by f (x) = x2. Then ( 1, 1) is open in r, but f (( 1, 1)) = [0, 1) is not an open set. (c) let e rn be a closed set. Let {xk}k n be a sequence with xk e for all k n, and assume limk xk = x.
