zero set of a topological space

Let X be a topological spaceMathworldPlanetmath and fC(X), the ring of continuous functions on X. The level setPlanetmathPlanetmath of f at r is the set f-1(r):={xXf(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 fC(X).

Properties. Let X be a topological space and, unless otherwise specified, fC(X).

  1. 1.

    Any zero set of X is closed. The converseMathworldPlanetmath is not true. However, if X is a metric space, then any closed setPlanetmathPlanetmath A is a zero set: simply define f:X by f(x):=d(x,A) where d is the metric on X.

  2. 2.

    The level set of f at r is the zero set of f-r^, where r^ is the constant function valued at r.

  3. 3.

    Z(r^)=X iff r=0. Otherwise, Z(r^)=. In fact, Z(f)= iff f is a unit in the ring C(X).

  4. 4.

    Since f(a)=0 iff |f(a)|<1n for all n, and each {xX|f(x)|<1n} is open in X, we see that


    This shows every zero set is a Gδ ( set.

  5. 5.

    For any fC(X), Z(f)=Z(fn)=Z(|f|), where n is any positive integer.

  6. 6.


  7. 7.


  8. 8.

    {xX0f(x)} is a zero set, since it is equal to Z(f-|f|).

  9. 9.

    If C(X) is considered as an algebra over , then Z(rf)=Z(f) iff r0.

The complementPlanetmathPlanetmath of a zero set is called a cozero set. In other words, a cozero set looks like {xXf(x)0} for some fC(X). By the last property above, a cozero set also has the form pos(f):={xX0<f(x)} for some fC(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)fA}. 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.


  • By properties 6. and 7. above, Z(X) is closed under set union and set intersectionMathworldPlanetmath operations. It can be shown that Z(X) is also closed under countableMathworldPlanetmath intersections.

  • It is also possible to define a zero set of X to be the zero set of some fC*(X), the subring of C(X) consisting of the boundedPlanetmathPlanetmathPlanetmathPlanetmath continuous functionsMathworldPlanetmathPlanetmath into . However, this definition turns out to be equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the one given for C(X), by the observation that Z(f)=Z(|f|1^).


  • 1 L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).
Title zero set of a topological space
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