Saturated set explained

In mathematics, particularly in the subfields of set theory and topology, a set

C is said to be saturated with respect to a function f : X \to Y if C is a subset of f 's domain X and if whenever f sends two points c \in C and x \in X to the same value then x belongs to C (that is, if f(x) = f(c) then x \in C ). Said more succinctly, the set C is called saturated if C = f ^{-1} (f(C)). (X, \tau) is saturated if it is equal to an intersection of open subsets of X. In a T₁ space every set is saturated.

Definition

Preliminaries

Let

f : X \to Y be a map. Given any subset S\subseteq X, define its under f to be the set:$$f(S)\; :=\; \backslash$$ and define its or under f to be the set:$$f^(S)\; :=\; \backslash .$$

Given

y \in Y, is defined to be the preimage:$$f^(y)\; :=\; f^(\backslash )\; =\; \backslash .$$

Any preimage of a single point in

f 's codomain Y is referred to as

Saturated sets

A set

C is called and is said to be if C is a subset of f 's domain X and if any of the following equivalent conditions are satisfied: C = f ^{-1} (f(C)).

1. There exists a set

S such that C = f ^{-1} (S).

1. • Any such set

S necessarily contains f(C) as a subset and moreover, it will also necessarily satisfy the equality f(C) = S \cap \operatorname{Im} f, where \operatorname{Im} f : = f(X) denotes the image of f.

1. If

c \in C and x \in X satisfy f(x) = f(c), then x \in C.

1. If

y \in Y is such that the fiber f ^{-1} (y) intersects C (that is, if f ^{-1} (y) \cap C ≠ \varnothing ), then this entire fiber is necessarily a subset of C (that is, f ^{-1} (y) \subseteq C ).

1. For every

y \in Y, the intersection C \cap f ^{-1} (y) is equal to the empty set \varnothing or to f ^{-1} (y). Related to computability theory, this notion can be extended to programs. Here, considering a subset A \subseteq N , this can be considered saturated (or extensional) if \forall m, n \in N , m \in A, \vee \phi _{m} = \phi _{n} ⇒ n \in A . 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

A be a subset such that A ≠ \emptyset, A ≠ N , A \subseteq N . If A is saturated, then A is not recursive.

Examples

Let

f : X \to Y be any function. If S is set then its preimage C: = f ^{-1} (S) under f is necessarily an f -saturated set. In particular, every fiber of a map f is an f -saturated set. \varnothing = f ^{-1} (\varnothing) and the domain X = f ^{-1} (Y) are always saturated. Arbitrary unions of saturated sets are saturated, as are arbitrary intersections of saturated sets.

Properties

Let

S and T be any sets and let f : X \to Y be any function.

If

S T is f -saturated then$$f(S\; \backslash cap\; T)\; ~=~\; f(S)\; \backslash cap\; f(T).$$

If

T is f -saturated then$$f(S\; \backslash setminus\; T)\; ~=~\; f(S)\; \backslash setminus\; f(T)$$where note, in particular, that requirements or conditions were placed on the set S.

If

\tau is a topology on X and f : X \to Y is any map then set \tau _{f} of all U \in \tau that are saturated subsets of X forms a topology on X. If Y is also a topological space then f : (X, \tau) \to Y is continuous (respectively, a quotient map) if and only if the same is true of f : \left(X, \tau _{f\right)} \to Y.

References

• Encyclopedia: G. Gierz. K. H. Hofmann. K. Keimel. J. D. Lawson. M. Mislove. D. S. Scott. Dana Scott. amp. Encyclopedia of Mathematics and its Applications. Continuous Lattices and Domains. 2003. Cambridge University Press. 93. 0-521-80338-1. registration.
• Book: Monk, James Donald. Introduction to Set Theory. McGraw-Hill. New York. 1969. International series in pure and applied mathematics. 978-0-07-042715-0. 1102.