In
general topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometri ...
, a branch of
mathematics, a non-empty family ''A'' of
subsets of a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
is said to have the finite intersection property (FIP) if the
intersection over any finite subcollection of
is
non-empty
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 t ...
. It has the strong finite intersection property (SFIP) if the intersection over any finite subcollection of
is infinite. Sets with the finite intersection property are also called centered systems and filter subbases.
The finite intersection property can be used to reformulate topological
compactness in terms of
closed sets; this is its most prominent application. Other applications include proving that certain
perfect sets are uncountable, and the construction of
ultrafilters.
Definition
Let
be a set and
a
nonempty
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 ...
family of subsets of that is,
is a
subset of the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
of Then
is said to have the finite intersection property if every nonempty finite subfamily has nonempty intersection; it is said to have the strong finite intersection property if that intersection is always infinite.
In symbols,
has the FIP if, for any choice of a finite nonempty subset
of there must exist a point
Likewise,
has the SFIP if, for every choice of such there are infinitely many such
In the study of
filters
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
, the common intersection of a family of sets is called a
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
, from much the same etymology as the
sunflower. Families with empty kernel are called
free; those with nonempty kernel,
fixed
Fixed may refer to:
* ''Fixed'' (EP), EP by Nine Inch Nails
* ''Fixed'', an upcoming 2D adult animated film directed by Genndy Tartakovsky
* Fixed (typeface), a collection of monospace bitmap fonts that is distributed with the X Window System
* ...
.
Families of examples and non-examples
The empty set cannot belong to any collection with the finite intersection property.
A sufficient condition for the FIP intersection property is a nonempty kernel. The converse is generally false, but holds for finite families; that is, if
is finite, then
has the finite intersection property if and only if it is fixed.
Pairwise intersection
The finite intersection property is ''strictly stronger'' than pairwise intersection; the family
has pairwise intersections, but not the FIP.
More generally, let
be a positive integer greater than unity, and Then any subset of
with fewer than
elements has nonempty intersection, but
lacks the FIP.
End-type constructions
If
is a decreasing sequence of non-empty sets, then the family
has the finite intersection property (and is even a
–system). If the inclusions
are
strict, then
admits the strong finite intersection property as well.
More generally, any
that is
totally ordered
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive ...
by inclusion has the FIP.
At the same time, the kernel of
may be empty: if then the
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
of
is the
empty set. Similarly, the family of intervals
also has the (S)FIP, but empty kernel.
"Generic" sets and properties
The family of all
Borel subsets of