zero set of a topological space
Let be a topological space and , the ring of continuous functions on . The level set of at is the set . The zero set of is defined to be the level set of at . The zero set of is denoted by . A subset of is called a zero set of if for some .
Properties. Let be a topological space and, unless otherwise specified, .
The level set of at is the zero set of , where is the constant function valued at .
iff . Otherwise, . In fact, iff is a unit in the ring .
Since iff for all , and each is open in , we see that
This shows every zero set is a (http://planetmath.org/G_deltaSet) set.
For any , , where is any positive integer.
is a zero set, since it is equal to .
If is considered as an algebra over , then iff .
The complement of a zero set is called a cozero set. In other words, a cozero set looks like for some . By the last property above, a cozero set also has the form for some .
Let be a subset of . The zero set of is defined as the set of all zero sets of elements of : . When , we also write and call it the family of zero sets of . Evidently, is a subset of the family of all closed sets of .
- 1 L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).
|Title||zero set of a topological space|
|Date of creation||2013-03-22 16:56:06|
|Last modified on||2013-03-22 16:56:06|
|Last modified by||CWoo (3771)|