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zero set of a topological space∗
CWoo†
2013-03-21 22:33:24
Let X be a topological space and f ∈ C(X), the ring of continuous functions
on X. The level set of f at r ∈ R is the set f −1 (r) := {x ∈ X | f (x) = r}.
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 f ∈ C(X).
Properties. Let X be a topological space and, unless otherwise specified,
f ∈ C(X).
1. Any zero set of X is closed. The converse is not true. However, if X is a
metric space, then any closed set A is a zero set: simply define f : X → R
by f (x) := d(x, A) where d is the metric on X.
2. The level set of f at r is the zero set of f − r̂, where r̂ is the constant
function valued at r.
3. Z(r̂) = X iff r = 0. Otherwise, Z(r̂) = ∅. In fact, Z(f ) = ∅ iff f is a
unit in the ring C(X).
4. Since f (a) = 0 iff |f (a)| <
is open in X, we see that
1
n
Z(f ) =
for all n ∈ N, and each {x ∈ X | |f (x)| <
∞
\
{x ∈ X | |f (x)| <
n=1
1
n}
1
}.
n
This shows every zero set is a Gδ set.
5. For any f ∈ C(X), Z(f ) = Z(f n ) = Z(|f |), where n is any positive
integer.
6. Z(f g) = Z(f ) ∪ Z(g).
7. Z(f ) ∩ Z(g) = Z(f 2 + g 2 ) = Z(|f | + |g|).
∗ hZeroSetOfATopologicalSpacei created: h2013-03-21i by: hCWooi version: h39201i
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8. {x ∈ X | 0 ≤ f (x)} is a zero set, since it is equal to Z(f − |f |).
9. If C(X) is considered as an algebra over R, then Z(rf ) = Z(f ) iff r 6= 0.
The complement of a zero set is called a cozero set. In other words, a cozero
set looks like {x ∈ X | f (x) 6= 0} for some f ∈ C(X). By the last property
above, a cozero set also has the form pos(f ) := {x ∈ X | 0 < f (x)} for some
f ∈ C(X).
Let A be a subset of C(X). The zero set of A is defined as the set of all zero
sets of elements of A: Z(A) := {Z(f ) | f ∈ A}. When A = C(X), we also write
Z(X) := Z(C(X)) and call it the family of zero sets of X. Evidently, Z(X) is
a subset of the family of all closed Gδ sets of X.
Remarks.
• By properties 6. and 7. above, Z(X) is closed under set union and set
intersection operations. It can be shown that Z(X) is also closed under
countable intersections.
• It is also possible to define a zero set of X to be the zero set of some
f ∈ C ∗ (X), the subring of C(X) consisting of the bounded continuous
functions into R. However, this definition turns out to be equivalent to
the one given for C(X), by the observation that Z(f ) = Z(|f | ∧ 1̂).
References
[1] L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand,
(1960).
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