• Support PF! Buy your school textbooks, materials and every day products Here!

Proof: if A is subset of B then closure of A is subset of closure of B

  • Thread starter chessbrah
  • Start date
  • #1
2
0
I'm thinking proof by contradiction but I can't seem to get anywhere.
 

Answers and Replies

  • #2
pwsnafu
Science Advisor
1,080
85
Think about limit points. x is in the closure of A, so it is a limit point of A...
 
  • #3
Fredrik
Staff Emeritus
Science Advisor
Gold Member
10,851
406
You should probably state the definition of "closure" that you would like to use, and maybe also your definition of "closed". I like the definition that says that the closure of E is the smallest closed set that contains E. If we use that definition, the proof is very simple.

I see that you're new here. If you're wondering why we don't just tell you the complete solution, it's because the forum rules tell us not to do that when someone posts a textbook-style question. We are only allowed to give you hints, and tell you if you're doing something wrong.
 
  • #4
disregardthat
Science Advisor
1,854
33
It might be easier to prove that if A is contained in a closed set (here the closure of B, which follows since B is contained in the closure of B and A is contained in B), then the closure of A is contained in the same closed set.
 
  • #5
PAllen
Science Advisor
2019 Award
7,915
1,200
May I ask why this thread is here? It isn't number theory, and it looks like homework to me.
 
  • #6
HallsofIvy
Science Advisor
Homework Helper
41,794
925
May I ask why this thread is here? It isn't number theory, and it looks like homework to me.
I agree. I have moved the thread to "Homework and Coursework- Calculus and Beyond".

I also agree with Fredrik that you need to state what definition of closure you are using. One common definition is that the closure of a set, A, is the smallest closed set that contains A. Using that definition this theorem is pretty close to trivial.
 

Related Threads for: Proof: if A is subset of B then closure of A is subset of closure of B

Replies
6
Views
12K
  • Last Post
Replies
4
Views
2K
Replies
1
Views
1K
Replies
3
Views
1K
Replies
2
Views
2K
Replies
1
Views
730
Replies
5
Views
16K
Top