Closure of { (x,sin(1/x) : 0<x<=1 }?

  • #1
filter54321
39
0

Homework Statement


What is the closure of F = { (x,sin(1/x) : 0<x<=1 }?


Homework Equations


None


The Attempt at a Solution


F is a squiggly line in R2. For every point in F (every point on the squiggly line) an open ball about that point will contain point both in F and in the complement of F. Therefore F is it's own boundary.

The closure of a set is equal to the unions of the boundary of the set and the set itself
Sclosure = dS U S

Therefore Fclosure = dF U F = F U F = F

My professor indicated that this line of reasoning is flawed. I'm not sure why nor am I sure what a correct anyswer would be. Any help would be appreciated.
 

Answers and Replies

  • #2
Mathnerdmo
41
0
What do you know about sin(1/x)?

Look at the graph again. It does something very interesting.
 
  • #3
filter54321
39
0
I know what it looks like, it's a common example in calculus. It bounces up and down as you go to 0 with ever increasing frequency. How does that have anything to do with the boundary methodology outlined in my original post?
 
  • #4
JG89
728
1
Use the fact that Cl(F) = lim(F) = {x : x is a limit of F}. Now let x approach any value in (0,1] and look at the set of F's limits.
 
  • #5
Mathnerdmo
41
0
The graph of y = sin(1/x), in particular that it oscillates with increasing frequency as x gets closer to 0, has everything to do with the boundary.

F is a squiggly line in R2. For every point in F (every point on the squiggly line) an open ball about that point will contain point both in F and in the complement of F. Therefore F is it's own boundary.

The problem is right here. All you've shown is that F is a subset of its boundary. There may be other points of R2 in the boundary.
 

Suggested for: Closure of { (x,sin(1/x) : 0<x<=1 }?

Replies
5
Views
355
  • Last Post
Replies
2
Views
67
Replies
2
Views
370
Replies
29
Views
649
Replies
16
Views
117
  • Last Post
Replies
14
Views
889
Replies
9
Views
649
  • Last Post
Replies
5
Views
182
  • Last Post
2
Replies
54
Views
2K
Top