mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, particularly in the subfields of

set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

and

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, a set

C

is said to be saturated with respect to a function

f : X \to Y

C

is a subset of

f

domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...

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^(f(C)).

, a

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

of a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

(X, \tau)

is saturated if it is equal to an

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...

open subsets In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a Set (mathematics), set along with a metric (mathematics), distance defined between any two points), open sets are the sets that, with every ...

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) := \

and define its or under

f

to be the set:

f^(S) := \.

Given

y \in Y,

is defined to be the preimage:

f^(y) := f^(\) = \.

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^(f(C)).

# There exists a set

S

such that

C = f^(S).

#* 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 f,

where

\operatorname f := f(X)

denotes the

image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...

f.

# If

c \in C

and

x \in X

satisfy

f(x) = f(c),

then

x \in C.

# If

y \in Y

is such that the fiber

f^(y)

intersects

C

(that is, if

f^(y) \cap C \neq \varnothing

), then this entire fiber is necessarily a subset of

C

(that is,

f^(y) \subseteq C

). # For every

y \in Y,

the intersection

C \cap f^(y)

is equal to the

empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...

\varnothing

or to

f^(y).

Examples

Let

f : X \to Y

be any function. If

S

is set then its preimage

C:= f^(S)

under

f

is necessarily an

f

-saturated set. In particular, every fiber of a map

f

is an

f

-saturated set. The

\varnothing = f^(\varnothing)

and the domain

X = f^(Y)

are always saturated. Arbitrary

union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...

s of saturated sets are saturated, as are arbitrary

s of saturated sets.

Properties

Let

S

and

T

be any sets and let

f : X \to Y

be any function. If

S

T

f

-saturated then

f(S \cap T) ~=~ f(S) \cap f(T).

T

f

-saturated then

f(S \setminus T) ~=~ f(S) \setminus f(T)

where note, in particular, that requirements or conditions were placed on the set

S.

\tau

is a

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.

Y

is also a topological space then

f : (X, \tau) \to Y

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...

(respectively, a

quotient map In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...

) if and only if the same is true of

f : \left(X, \tau_f\right) \to Y.

References