Hi,

How can I show that:

a) any zero set is closed

b) not every closed set is zero set?

Thanks for any help.

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- March 1st 2010, 10:05 AMArczi1984Zero set
Hi,

How can I show that:

a) any zero set is closed

b) not every closed set is zero set?

Thanks for any help. - March 1st 2010, 10:09 AMPlato
- March 1st 2010, 10:20 AMArczi1984
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, 10:47 AMfacenian
Let then then there exists such that and because of continuity there exist a neibourhood U_x of x such that ,ie,Z(f) is closed because its complement is open

- March 1st 2010, 11:09 AMArczi1984
Thanks.

I tried to solve the second part. I found some example but I do not understand the last part. Here is this example:

Let us consider X=Z and . Clearly is a closed subset but it is not a zero-set "because ".

This "" part in not clear for me. - March 1st 2010, 12:38 PMDrexel28
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.