# open? closed? both?

• Apr 10th 2010, 10:32 AM
swallenberg
open? closed? both?
Consider f(x) = 1/(1+x^2) as a function from R to R with the usual metric. Are the following statements true or false (give proofs or counterexamples):

1. f(A) is open for all open sets A in R
2. f(A) is open for all closed sets A in R

thanks!
• Apr 10th 2010, 10:40 AM
1. Let A=R. Then A is open and f(A) = (0,1].

2 is obviously false (take A to be a singleton). Did you mean f(A) is closed for closed sets A? If so, use the same example as (1).
• Apr 10th 2010, 10:47 AM
swallenberg
Quote:

1. Let A=R. Then A is open and f(A) = (0,1].

2 is obviously false (take A to be a singleton). Did you mean f(A) is closed for closed sets A? If so, use the same example as (1).

2. thanks!
nope... f(A) open if A is closed

1. I'm not sure that proves anything
• Apr 10th 2010, 10:48 AM
swallenberg
Quote:

1. Let A=R. Then A is open and f(A) = (0,1].

2 is obviously false (take A to be a singleton). Did you mean f(A) is closed for closed sets A? If so, use the same example as (1).

2. thanks!
nope... f(A) open if A is closed

1. I'm not sure that proves anything
• Apr 10th 2010, 10:49 AM
Is (0,1] open?
• Apr 10th 2010, 10:53 AM
swallenberg
Quote: