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

sends two points

Said more succinctly, the set

is called saturated if

is saturated if it is equal to an intersection of open subsets of

In a T1 space every set is saturated.

define its image under

and define its preimage or inverse image under

is defined to be the preimage:

Any preimage of a single point in

is referred to as a fiber of

-saturated and is said to be saturated with respect to

and if any of the following equivalent conditions are satisfied:[1] Related to computability theory, this notion can be extended to programs.

Here, considering a subset

, this can be considered saturated (or extensional) if

In words, given two programs, if the first program is in the set of programs satisfying the property and two programs are computing the same thing, then also the second program satisfies the property.

This means that if one program with a certain property is in the set, all programs computing the same function must also be in the set).

In this context, this notion can extend Rice's theorem, stating that: Let

is any set then its preimage

In particular, every fiber of a map

The empty set

Arbitrary unions of saturated sets are saturated, as are arbitrary intersections of saturated sets.

where note, in particular, that no requirements or conditions were placed on the set

is any map then set

forms a topology on

is also a topological space then

is continuous (respectively, a quotient map) if and only if the same is true of

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