EXAM 2 - MATH 470

EXAM 2 - MATH 470 open A set O contained in R is open if for all a in O there exists epsilon greater than 0 s.t. V_epsilon(a) is contained in O. Union and intersection theorem (i) The union of any arbitrary collection of open sets is an open set. (ii) The intersection of any finite collection of open sets is an open set. limit point A point x is a limit point of a set A if, for all epsilon >0, the set v_epsilon(x) n A contains a point other than x. limit point theorem A point x is a limit point of the set A iff x=lim(a_n) for some sequence (a_n) contained in A s.t. a_n/=x for all n in N. isolated If a in A, but it is not a limit point of A, then we call x an isolated point of A. closed A set F in R is closed if it contains all of its limit points. closed theorem A set F is closed iff every Cauchy sequence contained in F has its limit also in F. closure The closure of a set A is Au{all limit points of A} and is written A(bar). closure theorem A(bar) is always closed for any set A. closed set theorem If F is any closed set containing A, then F must also contain A(bar). complement theorem A set O is open iff O' is closed. A set F is closed iff F' is open. theorem of union and intersection of closed sets (i) The union of finitely many closed sets is a closed set. (ii) The intersection of any collection of closed sets is closed. compact A set K in R is compact is every sequence contained in K has a convergent subsequence whose limit is also in K. compact theorem A set K is compact iff K is closed and bounded. lim f(x)=L limit means for every epsilon >0 there is delta >0 s.t. for every x s.t. 0<|x-c|<delta, we have |f(x)-L|<epsilon Sequential Criterion theorem Let f:A-->R be a function and c a limit point of A. Then lim f(x) = L iff for all sequences (x_n) contained in A converging to c s.t. (x_n)/=c for all c, we h