In
general topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and related areas of
mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the final topology (or coinduced,
[
] strong, colimit, or inductive topology) on a
set with respect to a family of functions from
topological spaces
In mathematics, a topological space is, roughly speaking, a geometry, geometrical space in which ''closeness'' is defined but, generally, cannot be measured by a numeric distance. More specifically, a topological space is a Set (mathematics), set of ...
into
is the
finest topology on
that makes all 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 ga ...
.
The
quotient topology on a quotient space is a final topology, with respect to a single surjective function, namely the quotient map. The
disjoint union topology
In general topology
, a useful example in point-set topology. It is connected but not path-connected.
In mathematics, general topology is the branch of topology that deals with the basic Set theory, set-theoretic definitions and constructions used ...
is the final topology with respect to the
inclusion map
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s. The final topology is also the topology that every
direct limit
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
in the
category of topological spaces In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
is endowed with, and it is in the context of direct limits that the final topology often appears. A topology is Coherent topology, coherent with some collection of Subspace topology, subspaces if and only if it is the final topology induced by the natural inclusions.
The dual notion is the initial topology, which for a given family of functions from a set
into topological spaces is the coarsest topology on
that makes those functions continuous.
Definition
Given a set
and an
-indexed family of topological spaces
with associated functions
the is the
finest topology on
such that
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 ga ...
for each
.
Explicitly, the final topology may be described as follows:
:a subset
of
is open in the final topology
(that is,
) if and only if
is open in
for each
.
The closed subsets have an analogous characterization:
:a subset
of
is closed in the final topology
if and only if
is closed in
for each
.
Examples
The important special case where the family of maps
consists of a single surjective map can be completely characterized using the notion of Quotient map, quotient maps. A surjective function
between topological spaces is a quotient map if and only if the topology
on
coincides with the final topology
induced by the family
. In particular: the quotient topology is the final topology on the quotient space induced by the Quotient space (topology)#Quotient map, quotient map.
The final topology on a set
induced by a family of
-valued maps can be viewed as a far reaching generalization of the quotient topology, where multiple maps may be used instead of just one and where these maps are not required to be surjections.
Given topological spaces
, the disjoint union (topology), disjoint union topology on the disjoint union
is the final topology on the disjoint union induced by the natural injections.
Given a Family of sets, family of topologies
on a fixed set
the final topology on
with respect to the identity maps
as
ranges over
call it
is the infimum (or meet) of these topologies
in the lattice of topologies on
That is, the final topology
is equal to the Intersection (set theory), intersection
The
direct limit
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of any Direct system (mathematics), direct system of spaces and continuous maps is the set-theoretic direct limit together with the final topology determined by the canonical morphisms.
Explicitly, this means that if
is a direct system in the Category of topological spaces, category Top of topological spaces and if
is a
direct limit
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of
in the Category of sets, category Set of all sets, then by endowing
with the final topology
induced by
becomes the direct limit of
in the category Top.
The étalé space of a sheaf is topologized by a final topology.
A first-countable Hausdorff space
is locally path-connected if and only if
is equal to the final topology on
induced by the set
of all continuous maps
where any such map is called a Path (mathematics), path in
If a Hausdorff Locally convex topological vector space, locally convex topological vector space
is a Fréchet-Urysohn space then
is equal to the final topology on
induced by the set
of all Arc (topology), arcs in
which by definition are continuous Path (mathematics), paths
that are also topological embeddings.
Properties
Characterization via continuous maps
Given functions
from topological spaces
to the set
, the final topology on
can be characterized by the following property:
:a function
from
to some space
is continuous if and only if
is continuous for each
Behavior under composition
Suppose
is a family of maps, and for every
the topology
on
is the final topology induced by some family
of maps valued in
. Then the final topology on
induced by
is equal to the final topology on
induced by the maps
As a consequence: if
is the final topology on
induced by the family
and if
is any surjective map valued in some topological space
then
is a quotient map if and only if
has the final topology induced by the maps
By the universal property of the disjoint union topology we know that given any family of continuous maps
there is a unique continuous map
that is compatible with the natural injections.
If the family of maps
(i.e. each
lies in the image of some
) then the map
will be a quotient map if and only if
has the final topology induced by the maps
Effects of changing the family of maps
Throughout, let
be a family of
-valued maps with each map being of the form
and let
denote the final topology on
induced by
The definition of the final topology guarantees that for every index
the map
is continuous.
For any subset
the final topology
on
will be Comparison of topologies, than (and possibly equal to) the topology
; that is,
implies
where set equality might hold even if
is a proper subset of
If
is any topology on
such that for all
is continuous but
then
is Comparison of topologies, then
(in symbols,
which means
and
) and moreover, for any subset
because
the topology
will also be than the final topology
induced on
by
that is
Suppose that in addition,
is a family of
-valued maps whose domains are topological spaces
If every
is continuous then adding these maps to the family
will change the final topology on
that is,
Explicitly, this means that the final topology on
induced by the "extended family"
is equal to the final topology
induced by the original family
However, had there instead existed even just one map
such that
was continuous, then the final topology
on
induced by the "extended family"
would necessarily be Comparison of topologies, than the final topology
induced by
that is,
(see this footnote
[By definition, the map not being continuous means that there exists at least one open set such that is not open in In contrast, by definition of the final topology the map be continuous. So the reason why must be strictly coarser, rather than strictly finer, than is because the failure of the map to be continuous necessitates that one or more open subsets of must be "removed" in order for to become continuous. Thus is just but some open sets "removed" from ] for an explanation).
Coherence with subspaces
Let
be a topological space and let
be a Family of sets, family of subspaces of
where importantly, the word "sub" is used to indicate that each subset
is endowed with the subspace topology
inherited from
The space
is said to be with the family
of subspaces if
where
denotes the final topology induced by the
inclusion map
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s
where for every
the inclusion map takes the form
:
Unraveling the definition,
is coherent with
if and only if the following statement is true:
:for every subset
is open in
if and only if for every
is open in the Subspace topology, subspace
Closed sets can be checked instead:
is coherent with
if and only if for every subset
is closed in
if and only if for every
is closed in
For example, if
is a cover of a topological space
by open subspaces (i.e. open subsets of
endowed with the subspace topology) then
is coherent with
In contrast, if
is the set of all Singleton set, singleton subsets of
(each set being endowed with its unique topology) then
is coherent with
if and only if
is the discrete topology on
The Disjoint union (topology), disjoint union is the final topology with respect to the family of canonical injections.
A space
is called and a if
is coherent with the set
of all compact subspaces of
All first-countable spaces and all Hausdorff space, Hausdorff locally compact spaces are -spaces, so that in particular, every manifold and every metrizable space is coherent with the family of all its compact subspaces.
As demonstrated by the examples that follows, under certain circumstance, it may be possible to characterize a more general final topology in terms of coherence with subspaces.
Let
be a family of
-valued maps with each map being of the form
and let
denote the final topology on
induced by
Suppose that
is a topology on
and for every index
the Image (mathematics), image
is endowed with the subspace topology
inherited from
If for every
the map
is a quotient map then
if and only if
is coherent with the set of all images
Final topology on the direct limit of finite-dimensional Euclidean spaces
Let
denote the , where
denotes the space of all real sequences.
For every natural number
let
denote the usual Euclidean space endowed with the Euclidean topology and let
denote the
inclusion map
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
defined by
so that its Image (mathematics), image is
and consequently,
Endow the set
with the final topology
induced by the family
of all inclusion maps.
With this topology,
becomes a Complete topological vector space, complete Hausdorff space, Hausdorff Locally convex topological vector space, locally convex Sequential space, sequential topological vector space that is a Fréchet–Urysohn space.
The topology
is Comparison of topologies, strictly finer than the subspace topology induced on
by
where
is endowed with its usual product topology.
Endow the image
with the final topology induced on it by the bijection
that is, it is endowed with the Euclidean topology transferred to it from
via
This topology on
is equal to the subspace topology induced on it by
A subset
is open (resp. closed) in
if and only if for every
the set
is an open (resp. closed) subset of
The topology
is coherent with family of subspaces
This makes
into an LB-space.
Consequently, if
and
is a sequence in
then
in
if and only if there exists some
such that both
and
are contained in
and
in
Often, for every
the inclusion map
is used to identify
with its image
in
explicitly, the elements
and
are identified together.
Under this identification,
becomes a
direct limit
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of the direct system
where for every
the map
is the inclusion map defined by
where there are
trailing zeros.
Categorical description
In the language of category theory, the final topology construction can be described as follows. Let
be a functor from a discrete category
to the
category of topological spaces In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
Top that selects the spaces
for
Let
be the diagonal functor from Top to the functor category Top
''J'' (this functor sends each space
to the constant functor to
). The comma category
is then the category of co-cones from
i.e. objects in
are pairs
where
is a family of continuous maps to
If
is the forgetful functor from Top to Set and Δ′ is the diagonal functor from Set to Set
''J'' then the comma category
is the category of all co-cones from
The final topology construction can then be described as a functor from
to
This functor is Adjoint functors, left adjoint to the corresponding forgetful functor.
See also
*
*
*
*
*
Notes
Citations
References
* . ''(Provides a short, general introduction in section 9 and Exercise 9H)''
*
General topology