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, .
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1.
Any zero set of is closed. The converse is not true. However, if is a metric space, then any closed set is a zero set: simply define by where is the metric on .
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2.
The level set of at is the zero set of , where is the constant function valued at .
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3.
iff . Otherwise, . In fact, iff is a unit in the ring .
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4.
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.
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5.
For any , , where is any positive integer.
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6.
.
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7.
.
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8.
is a zero set, since it is equal to .
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9.
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 .
Remarks.
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By properties 6. and 7. above, is closed under set union and set intersection operations. It can be shown that is also closed under countable intersections.
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It is also possible to define a zero set of to be the zero set of some , the subring of consisting of the bounded continuous functions into . However, this definition turns out to be equivalent to the one given for , by the observation that .
References
- 1 L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).
Title | zero set of a topological space |
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Canonical name | ZeroSetOfATopologicalSpace |
Date of creation | 2013-03-22 16:56:06 |
Last modified on | 2013-03-22 16:56:06 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 10 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 54C50 |
Classification | msc 54C40 |
Classification | msc 54C35 |
Defines | zero set |
Defines | level set |
Defines | cozero set |