# Zero set

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In mathematics, the **zero set** of a real-valued function *f* : *X* → **R** (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* : *X* → **R** 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 **R**^{p} to **R**^{n}. 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 **R**^{m+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 **R**^{n} is the zero set of a smooth function defined on all of **R**^{n}. 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.