View source for Saturated set

In [[mathematics]], particularly in the subfields of [[set theory]] and [[topology]], a set <math>C</math> is said to be '''saturated''' with respect to a function <math>f : X \to Y</math> if <math>C</math> is a subset of <math>f</math>'s [[Domain of a function|domain]] <math>X</math> and if whenever <math>f</math> sends two points <math>c \in C</math> and <math>x \in X</math> to the same value then <math>x</math> belongs to <math>C</math> (that is, if <math>f(x) = f(c)</math> then <math>x \in C</math>). Said more succinctly, the set <math>C</math> is called saturated if <math>C = f^{-1}(f(C)).</math> 

In [[topology]], a [[subset]] of a [[topological space]] <math>(X, \tau)</math> is '''saturated''' if it is equal to an [[Intersection (set theory)|intersection]] of [[Open set|open subsets]] of <math>X.</math> In a [[T1 space|T<sub>1</sub> space]] every set is saturated. 

==Definition==

===Preliminaries===

Let <math>f : X \to Y</math> be a map. 
Given any subset <math>S\subseteq X,</math> define its {{em|[[Image (mathematics)|image]]}} under <math>f</math> to be the set:
<math display=block>f(S) := \{f(s) ~:~ s \in S\}</math> 
and define its {{em|[[preimage]]}} or {{em|[[inverse image]]}} under <math>f</math> to be the set:
<math display=block>f^{-1}(S) := \{x \in X ~:~ f(x) \in S\}.</math>

Given <math>y \in Y,</math> {{em|the [[Fiber (mathematics)|fiber]] of <math>f</math> over <math>y</math>}} is defined to be the preimage:
<math display=block>f^{-1}(y) := f^{-1}(\{y\}) = \{x \in X ~:~ f(x) = y\}.</math>

Any preimage of a single point in <math>f</math>'s codomain <math>Y</math> is referred to as {{em|'''a''' fiber of <math>f.</math>}}

===Saturated sets===

A set <math>C</math> is called {{em|'''<math>f</math>-saturated'''}} and is said to be {{em|'''saturated''' with respect to <math>f</math>}} if <math>C</math> is a subset of <math>f</math>'s domain <math>X</math> and if any of the following equivalent conditions are satisfied:{{sfn|Monk|1969|pp=24-54}} 

# <math>C = f^{-1}(f(C)).</math>
# There exists a set <math>S</math> such that <math>C = f^{-1}(S).</math>
#* Any such set <math>S</math> necessarily contains <math>f(C)</math> as a subset and moreover, it will also necessarily satisfy the equality <math>f(C) = S \cap \operatorname{Im} f,</math> where <math>\operatorname{Im} f := f(X)</math> denotes the [[Image (mathematics)|image]] of <math>f.</math>
# If <math>c \in C</math> and <math>x \in X</math> satisfy <math>f(x) = f(c),</math> then <math>x \in C.</math>
# If <math>y \in Y</math> is such that the fiber <math>f^{-1}(y)</math> intersects <math>C</math> (that is, if <math>f^{-1}(y) \cap C \neq \varnothing</math>), then this entire fiber is necessarily a subset of <math>C</math> (that is, <math>f^{-1}(y) \subseteq C</math>).
# For every <math>y \in Y,</math> the intersection <math>C \cap f^{-1}(y)</math> is equal to the [[empty set]] <math>\varnothing</math> or to <math>f^{-1}(y).</math>
Related to [[computability theory]], this notion can be extended to programs. Here, considering a subset <math>A \subseteq \mathbb{N}</math>, this can be considered '''saturated''' (or '''extensional''') if <math>\forall m, n \in \mathbb{N}, m \in A, \vee \phi_m = \phi_n \Rightarrow n \in A</math>. 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 <math>A</math> be a subset such that <math>A \neq \emptyset, A \neq \mathbb{N}, A \subseteq N </math>. If <math>A</math> is saturated, then <math>A</math> is not recursive.

==Examples==

Let <math>f : X \to Y</math> be any function. If <math>S</math> is {{em|any}} set then its preimage <math>C:= f^{-1}(S)</math> under <math>f</math> is necessarily an <math>f</math>-saturated set. In particular, every fiber of a map <math>f</math> is an <math>f</math>-saturated set. 

The [[empty set]] <math>\varnothing = f^{-1}(\varnothing)</math> and the domain <math>X = f^{-1}(Y)</math> are always saturated. Arbitrary [[union (mathematics)|union]]s of saturated sets are saturated, as are arbitrary [[intersection]]s of saturated sets. 

==Properties==

Let <math>S</math> and <math>T</math> be any sets and let <math>f : X \to Y</math> be any function. 

If <math>S</math> '''{{em|or}}''' <math>T</math> is <math>f</math>-saturated then
<math display=block>f(S \cap T) ~=~ f(S) \cap f(T).</math>

If <math>T</math> is <math>f</math>-saturated then
<math display=block>f(S \setminus T) ~=~ f(S) \setminus f(T)</math>
where note, in particular, that {{em|no}} requirements or conditions were placed on the set <math>S.</math>

If <math>\tau</math> is a [[Topology (structure)|topology]] on <math>X</math> and <math>f : X \to Y</math> is any map then set <math>\tau_f</math> of all <math>U \in \tau</math> that are saturated subsets of <math>X</math> forms a topology on <math>X.</math> If <math>Y</math> is also a topological space then <math>f : (X, \tau) \to Y</math> is [[Continuous function (topology)|continuous]] (respectively, a [[quotient map (topology)|quotient map]]) if and only if the same is true of <math>f : \left(X, \tau_f\right) \to Y.</math>

==See also==

* {{annotated link|List of set identities and relations}}

==References==
{{reflist}}

* {{cite encyclopedia|author1=G. Gierz|author2=K. H. Hofmann|author3=K. Keimel|author4=J. D. Lawson|author5=M. Mislove|author6=D. S. Scott|author6link = Dana Scott|name-list-style=amp|encyclopedia=Encyclopedia of Mathematics and its Applications|title=Continuous Lattices and Domains|year=2003|publisher=Cambridge University Press|volume=93|isbn=0-521-80338-1|url-access=registration|url=https://archive.org/details/continuouslattic0000unse}}
* {{cite book|last=Monk|first=James Donald|title=Introduction to Set Theory|publisher=McGraw-Hill|publication-place=New York|url=http://euclid.colorado.edu/~monkd/monk11.pdf|year=1969|series=International series in pure and applied mathematics|isbn=978-0-07-042715-0|oclc=1102}} <!-- {{sfn | Monk | 1969 | p=}} -->
* {{Munkres Topology|edition=2}} <!-- {{sfn | Munkres | 2000 | p=}} -->

{{Mathematical logic}}

[[Category:Basic concepts in set theory]]
[[Category:General topology]]
[[Category:Operations on sets]]

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