support of function

Definition Suppose X is a topological spaceMathworldPlanetmath, and f:X is a function. Then the support of f (written as suppf), is the set


In other words, suppf is the closurePlanetmathPlanetmath of the set where f does not vanish.


Let f:X be a function.

  1. 1.

    suppf is closed.

  2. 2.

    If xsuppf, then f(x)=0.

  3. 3.

    If suppf=, then f=0.

  4. 4.

    If χ:X is such that χ=1 on suppf, then f=χf.

  5. 5.

    If f,g:X are functions, then we have

    supp(fg) suppfsuppg,
    supp(f+g) suppfsuppg.
  6. 6.

    If Y is another topological space, and Ψ:YX is a homeomorphismPlanetmathPlanetmath, then

Title support of function
Canonical name SupportOfFunction
Date of creation 2013-03-22 13:46:10
Last modified on 2013-03-22 13:46:10
Owner matte (1858)
Last modified by matte (1858)
Numerical id 16
Author matte (1858)
Entry type Definition
Classification msc 54-00
Synonym support
Synonym carrier
Related topic ZeroOfAFunction
Related topic ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces