Saturated set
In mathematics, particularly in the subfields of set theory and topology, a set formula_1 is said to be saturated with respect to a function formula_2 if formula_1 is a subset of formula_4's domain formula_5 and if whenever formula_4 sends two points formula_7 and formula_8 to the same value then formula_9 belongs to formula_1 (that is, if formula_11 then formula_12). Said more succinctly, the set formula_1 is called saturated if formula_14 In topology, a subset of a topological space formula_15 is saturated if it is equal to an intersection of open subsets of formula_16 In a T space  every set is saturated. Definition. Preliminaries. Let formula_2 be a map. Given any set formula_18 define its under formula_4 to be the set: formula_20 and define its or under formula_4 to be the set: formula_22 Given formula_23 is defined to be the preimage: formula_24 Any preimage of a single point in formula_4's codomain formula_26 is referred to as . Saturated sets. A set formula_1 is called and is said to be if formula_1 be a subset of formula_4's domain formula_5 and if any of the following equivalent conditions are satisfied: Examples. Let formula_2 be any function. If formula_32 is  set then its preimage formula_55 under formula_4 is necessarily an formula_4-saturated set. In particular, every fiber of a map formula_4 is an formula_4-saturated set. Properties. Let formula_32 and formula_61 be any sets and let formula_2 be any function. If formula_32 formula_61 is formula_4-saturated then formula_66 If formula_61 is formula_4-saturated then formula_69 where note, in particular, that requirements or conditions were placed on the set formula_70