Zero set
In mathematics, and more specifically in geometry and topology, 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 consisting of all elements x satisfying f(x) = 0. 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).
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 every cozero set is an open set, but the converses do not always hold. In fact:
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.
...
Wikipedia