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 collecti...
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.
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