Zero Set - 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).