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.
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller CertifiedGrades. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for £8.30. You're not tied to anything after your purchase.
