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zero set of a topological space
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(Definition)
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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 0. 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, .
- 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 .
- 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
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 .
Remarks.
- 1
- L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).
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"zero set of a topological space" is owned by CWoo.
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(view preamble)
| Also defines: |
zero set, level set, cozero set |
This object's parent.
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Cross-references: equivalent, continuous functions, bounded, subring, countable, operations, intersection, union, closed under, complement, algebra, integer, positive, open, ring, unit, iff, constant function, metric, closed set, metric space, converse, closed, properties, subset, ring of continuous functions, topological space
There are 8 references to this entry.
This is version 7 of zero set of a topological space, born on 2007-04-16, modified 2007-05-14.
Object id is 9201, canonical name is ZeroSetOfATopologicalSpace.
Accessed 1621 times total.
Classification:
| AMS MSC: | 54C35 (General topology :: Maps and general types of spaces defined by maps :: Function spaces) | | | 54C40 (General topology :: Maps and general types of spaces defined by maps :: Algebraic properties of function spaces) | | | 54C50 (General topology :: Maps and general types of spaces defined by maps :: Special sets defined by functions) |
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Pending Errata and Addenda
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