Saturated set

In mathematics, particularly in the subfields of set theory and topology, a set is said to be saturated with respect to a function if is a subset of 's domain and if whenever sends two points and to the same value then belongs to (that is, if then ). Said more succinctly, the set is called saturated if In topology, a subset of a topological space is saturated if it is equal to an intersection of open subsets of In a T1 space every set is saturated. Let be a map. Given any subset define its under to be the set: and define its or under to be the set: Given is defined to be the preimage: Any preimage of a single point in 's codomain is referred to as A set is called and is said to be if is a subset of 's domain and if any of the following equivalent conditions are satisfied: There exists a set such that Any such set necessarily contains as a subset and moreover, it will also necessarily satisfy the equality where denotes the of If and satisfy then If is such that the fiber intersects (that is, if ), then this entire fiber is necessarily a subset of (that is, ). For every the intersection is equal to the empty set or to Let be any function. If is set then its preimage under is necessarily an -saturated set. In particular, every fiber of a map is an -saturated set. The empty set and the domain are always saturated. Arbitrary unions of saturated sets are saturated, as are arbitrary intersections of saturated sets. Let and be any sets and let be any function.