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 . 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.
Definition
Preliminaries
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
Saturated sets
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 image of
- If and satisfy then
- If is such that the fiber intersects , then this entire fiber is necessarily a subset of .
- For every the intersection is equal to the empty set or to
In this context, this notion can extend Rice's theorem, stating that:
Let be a subset such that. If is saturated, then is not recursive.
Examples
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.
Properties
Let and be any sets and let be any function.If is -saturated then
If is -saturated then
where note, in particular, that requirements or conditions were placed on the set
If is a topology on and is any map then set of all that are -saturated subsets of forms a topology on If is also a topological space then is continuous if and only if the same is true of