Zero set

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In mathematics, the zero set of a real-valued function f : XR (or more generally, a function taking values in some additive group) is the subset of X (the inverse image of {0}). In other words, the zero set of the function f is the subset of X on which . The cozero set of f is the complement of the zero set of f (i.e. the subset of X on which f is nonzero).

Zero sets are important in several branches of geometry and topology.

Topology

In topology, zero sets are defined with respect to continuous functions. Let X be a topological space, and let A be a subset of X. Then A is a zero set in X if there exists a continuous function f : XR such that

A cozero set in X is a subset whose complement is a zero set.

Every zero set is a closed set and a cozero set is an open set, but the converses does not always hold. In fact:

  • A topological space X is completely regular if and only if every closed set is the intersection of a family of zero sets in X. Equivalently, X is completely regular if and only if the cozero sets form a basis for X.
  • A topological space is perfectly normal if and only if every closed set is a zero set (equivalently, every open set is a cozero set).

Differential geometry

In differential geometry, zero sets are frequently used to define manifolds. An important special case is the case that f is a smooth function from Rp to Rn. If zero is a regular value of f then the zero-set of f is a smooth manifold of dimension m=p-n by the regular value theorem.

For example, the unit m-sphere in Rm+1 is the zero set of the real-valued function f(x) = |x|2 - 1.

An unrelated but important result in analysis and geometry states that any closet subset of Rn is the zero set of a smooth function defined on all of Rn. In fact, this result extends to any smooth manifold, as a corollary of paracompactness.

Algebraic geometry

In algebraic geometry, an affine variety is the zero set of a polynomial, or collection of polynomials. Similarly, a projective variety is the projectivization of the zero set of a collection of homogeneous polynomials.

References

Weisstein, Eric W., "Zero Set", MathWorld.