In
general topology and related areas of
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 ...
, the initial topology (or induced topology
or weak topology or limit topology or projective topology) on 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 ...
with respect to a family of functions on
is the
coarsest topology on ''X'' that makes those functions
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 g ...
.
The
subspace topology and
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...
constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these.
The
dual notion is the
final topology, which for a given family of functions mapping to a set
is the
finest topology
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies.
Definition
A topology on a set may be defined as th ...
on
that makes those functions continuous.
Definition
Given a set
and an
indexed family of
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 poin ...
s with functions
the initial topology
on
is the
coarsest topology on
such that each
is
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 g ...
.
Definition in terms of open sets
If
is a family of topologies
indexed by
then the of these topologies is the coarsest topology on
that is finer than each
This topology always exists and it is equal to the
topology generated by
If for every
denotes the topology
then
is a topology on
and the is the least upper bound topology of the
-indexed family of topologies
(for
).
Explicitly, the initial topology is the collection of open sets
generated by all sets of the form
where
is an
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
in
for some
under finite intersections and arbitrary unions.
Sets of the form
are often called . If
contains
exactly one element, then all the open sets of the initial topology
are cylinder sets.
Examples
Several topological constructions can be regarded as special cases of the initial topology.
* The
subspace topology is the initial topology on the subspace with respect to the
inclusion map.
* The
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...
is the initial topology with respect to the family of
projection maps.
* The
inverse limit of any
inverse system of spaces and continuous maps is the set-theoretic inverse limit together with the initial topology determined by the canonical morphisms.
* The
weak topology on a
locally convex space is the initial topology with respect to the
continuous linear form In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
An operator between two normed spaces is a bounded linear o ...
s of its
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
.
* Given a
family
Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Idea ...
of topologies
on a fixed set
the initial topology on
with respect to the functions
is the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
(or join) of the topologies
in the
lattice of topologies on
That is, the initial topology
is the topology generated by the
union of the topologies
* A topological space is
completely regular if and only if it has the initial topology with respect to its family of (
bounded) real-valued continuous functions.
* Every topological space
has the initial topology with respect to the family of continuous functions from
to the
Sierpiński space.
Properties
Characteristic property
The initial topology on
can be characterized by the following characteristic property:
A function
from some space
to
is continuous if and only if
is continuous for each
Note that, despite looking quite similar, this is not a universal property. A categorical description is given below.
A
filter on
converges to a point
if and only if the
prefilter converges to for every
Evaluation
By the universal property of the
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...
, we know that any family of continuous maps
determines a unique continuous map
This map is known as the .
A family of maps
is said to ''
'' in
if for all
in
there exists some
such that
The family
separates points if and only if the associated evaluation map
is
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
.
The evaluation map
will be a
topological embedding if and only if
has the initial topology determined by the maps
and this family of maps separates points in
Hausdorffness
If
has the initial topology induced by
and if every
is Hausdorff, then
is a
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...
if and only if these maps
separate points on
Transitivity of the initial topology
If
has the initial topology induced by the
-indexed family of mappings
and if for ever
the topology on
is the initial topology induced by some
-indexed family of mappings
(as
ranges over
), then the initial topology on
induced by
is equal to the initial topology induced by the
-indexed family of mappings
as
ranges over
and
ranges over
Several important corollaries of this fact are now given.
In particular, if
then the subspace topology that
inherits from
is equal to the initial topology induced by the
inclusion map (defined by
). Consequently, if
has the initial topology induced by
then the subspace topology that
inherits from
is equal to the initial topology induced on
by the restrictions
of the
to
The
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...
on
is equal to the initial topology induced by the canonical projections
as
ranges over
Consequently, the initial topology on
induced by
is equal to the inverse image of the product topology on
by the
evaluation map
In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' th ...
Furthermore, if the maps
separate points on
then the evaluation map is a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
onto the subspace
of the product space
Separating points from closed sets
If a space
comes equipped with a topology, it is often useful to know whether or not the topology on
is the initial topology induced by some family of maps on
This section gives a sufficient (but not necessary) condition.
A family of maps
''separates points from closed sets'' in
if for all
closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
s
in
and all
there exists some
such that
where
denotes the
closure operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S
:
Closure operators are de ...
.
:Theorem. A family of continuous maps
separates points from closed sets if and only if the cylinder sets
for
open in
form a
base for the topology on
It follows that whenever
separates points from closed sets, the space
has the initial topology induced by the maps
The converse fails, since generally the cylinder sets will only form a subbase (and not a base) for the initial topology.
If the space
is a
T0 space, then any collection of maps
that separates points from closed sets in
must also separate points. In this case, the evaluation map will be an embedding.
Initial uniform structure
If
is a family of
uniform structures on
indexed by
then the of
is the coarsest uniform structure on
that is finer than each
This uniform always exists and it is equal to the
filter on
generated by the
filter subbase
If
is the topology on
induced by the uniform structure
then the topology on
associated with least upper bound uniform structure is equal to the least upper bound topology of
Now suppose that
is a family of maps and for every
let
be a uniform structure on
Then the is the unique coarsest uniform structure
on
making all
uniformly continuous. It is equal to the least upper bound uniform structure of the
-indexed family of uniform structures
(for
).
The topology on
induced by
is the coarsest topology on
such that every
is continuous.
The initial uniform structure
is also equal to the coarsest uniform structure such that the identity mappings
are uniformly continuous.
Hausdorffness: The topology on
induced by the initial uniform structure
is
Hausdorff if and only if for whenever
are distinct (
) then there exists some
and some entourage
of
such that
Furthermore, if for every index
the topology on
induced by
is Hausdorff then the topology on
induced by the initial uniform structure
is Hausdorff if and only if the maps
separate points on
(or equivalently, if and only if the
evaluation map
In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' th ...
is injective)
Uniform continuity: If
is the initial uniform structure induced by the mappings
then a function
from some uniform space
into
is
uniformly continuous if and only if
is uniformly continuous for each
Cauchy filter: A
filter on
is a
Cauchy filter
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unif ...
on
if and only if
is a Cauchy prefilter on
for every
Transitivity of the initial uniform structure: If the word "topology" is replaced with "uniform structure" in the statement of "
transitivity of the initial topology" given above, then the resulting statement will also be true.
Categorical description
In the language of
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, the initial topology construction can be described as follows. Let
be the
functor from a
discrete category to the
category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again cont ...
which maps
. Let
be the usual
forgetful functor from
to
. The maps
can then be thought of as a
cone from
to
That is,
is an object of
—the
category of cones
In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well.
Definition
Let ''F'' : ''J'' → ''C'' be a diagram in ' ...
to
More precisely, this cone
defines a
-structured cosink in
The forgetful functor
induces a functor
. The characteristic property of the initial topology is equivalent to the statement that there exists a
universal morphism
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently f ...
from
to
that is, a terminal object in the category
Explicitly, this consists of an object
in
together with a morphism
such that for any object
in
and morphism
there exists a unique morphism
such that the following diagram commutes:
The assignment
placing the initial topology on
extends to a functor
which is
right adjoint to the forgetful functor
In fact,
is a right-inverse to
; since
is the identity functor on
See also
*
*
*
*
References
Bibliography
*
*
*
*
*
*
External links
*
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{{Topology, expanded
General topology