A community for students.
Here's the question you clicked on:
 0 viewing
anonymous
 4 years ago
"Every finite set is closed."
Is this true?
anonymous
 4 years ago
"Every finite set is closed." Is this true?

This Question is Closed

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Yes, because for a finite value, there is a definite number which is the last one, so you can close it.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0This is what I have: Proof: Let \(A\) be a finite set. Then \(A=\{x_1,x_2,...,x_n\}\) for some \(n\in\mathbb{N}\). Since every point in \(A\) is an isolated point. Therefore, \(A'=\varnothing\), and since \(\varnothing\in A\), it follows that \(A\) contains all its accumulation points. Therefore, \(A\) must be closed. \(\blacksquare\)

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.3You mean in R^n, under the usual topology? Yes. Now, in your proof, there's something odd going on here. What is A', the compliment of A?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0It's the set of all its accumulation points.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0I have a theorem that states that if a set contains all its accumulation points, then it must be closed.

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.3Ok, and then you mean to write \( \emptyset \subset A \). This is slightly obscure proof; I would use the definition of open and closed directly, as opposed to relying on such a result that comes farther down the road.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Another hint for a direct proof: any FINITE union of closed sets is closed. (You can construct open intervals in the usual topology on R by a countably infinite intersection of closed sets  you may want to prove that...).
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.