Results 1 to 6 of 6

- March 1st 2010, 11:05 AM #1

- Joined
- Oct 2009
- Posts
- 70

- March 1st 2010, 11:09 AM #2

- March 1st 2010, 11:20 AM #3

- Joined
- Oct 2009
- Posts
- 70

Here is definition which I used:

"Let X be a topological space and , the ring of continuous function on X . The level set of f is defined at is the set . The zero set of f is defined to be the level set of f at 0 . The zero set of f is denoted by Z(f) . A subset A of X is called a zero set of X if A=Z(f) for some .

- March 1st 2010, 11:47 AM #4

- Joined
- Feb 2010
- Posts
- 37

- March 1st 2010, 12:09 PM #5

- Joined
- Oct 2009
- Posts
- 70

- March 1st 2010, 01:38 PM #6
How about this. If is continuous (where is a metric space) we define the zero set . Why is closed? Because, is closed in under the usual topology and since is continuous so is

Alternatively, let . If we're done, so assume not. Then, is a limit point of and so there exists a sequence of points such that . But, since is continuous we know that and so . It follows that which finishes the argument.

What about which is the identity function. is closed in the domain but it's not the zero set.