Final topology

In general topology and related areas of mathematics, the final topology (or coinduced,[1] strong, colimit, or inductive topology) on a set with respect to a family of functions from topological spaces into is the finest topology on that makes all those functions continuous.

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 is the final topology with respect to the canonical inclusions. The final topology is also the topology that every direct limit in the category of topological spaces is endowed with, and it is in the context of direct limits that the final topology often appears. A topology is coherent with some collection of 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.

DefinitionEdit

Given a set   and an  -indexed family of topological spaces   with associated functions

 

the final topology on   induced by these maps   is the finest topology   on   such that

 

is continuous 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  .

ExamplesEdit

The important special case where the family of maps   consists of a single surjective map can be completely characterized using the notion of 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 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 on the disjoint union   is the final topology on the disjoint union induced by the natural injections.

Given a 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  

The direct limit of any 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 Top of topological spaces and if   is a direct limit of   in the 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 in  

If a Hausdorff locally convex topological vector space   is a Fréchet-Urysohn space then   is equal to the final topology on   induced by the set   of all arcs in   which by definition are continuous paths   that are also topological embeddings.

PropertiesEdit

Characterization via continuous mapsEdit

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 compositionEdit

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   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   covers   (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 mapsEdit

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 finer than (and possibly equal to) the final topology   induced on   by   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   is not equal to the final topology   induced on   by   then   is strictly coarser then   (i.e.   where in particular  ) and moreover, for any subset   because   the topology   will also be strictly coarser 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 not 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 not continuous, then the final topology   on   induced by the "extended family"   would necessarily be strictly coarser than the final topology   induced by   that is,   (see this footnote[note 1] for an explanation).

Coherence with subspacesEdit

Let   be a topological space and let   be a family of subspaces of   where importantly, the word "subspace" is used to indicate that each subset   is endowed with the subspace topology   inherited from   The space   is said to be coherent with the family   of subspaces if   where   denotes the final topology induced by the inclusion maps   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  

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 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 is the final topology with respect to the family of canonical injections. A space   is called compactly generated and a k-space if   is coherent with the set   of all compact subspaces of   All first-countable spaces and all Hausdorff locally compact spaces are k-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   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 spacesEdit

Let

 

denote the space of finite sequences, 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 canonical inclusion defined by   so that its image is

 

and consequently,

 

Endow the set   with the final topology   induced by the family   of all canonical inclusions. With this topology,   becomes a complete Hausdorff locally convex sequential topological vector space that is not a Fréchet–Urysohn space. The topology   is 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 canonical inclusion   is used to identify   with its image   in   explicitly, the elements   and   are identified together. Under this identification,   becomes a direct limit of the direct system   where for every   the map   is the canonical inclusion defined by   where there are   trailing zeros.

Categorical descriptionEdit

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 Top that selects the spaces   for   Let Δ be the diagonal functor from Top to the functor category TopJ (this functor sends each space   to the constant functor to  ). The comma category (Y ↓ Δ) is then the category of co-cones from   i.e. objects in (Y ↓ Δ) are pairs (X, f) 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 SetJ then the comma category (UY ↓ Δ′) is the category of all co-cones from   The final topology construction can then be described as a functor from (UY ↓ Δ′) to (Y ↓ Δ). This functor is left adjoint to the corresponding forgetful functor.

See alsoEdit

NotesEdit

  1. ^ 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   must 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  

CitationsEdit

  1. ^ Singh, Tej Bahadur (May 5, 2013). "Elements of Topology". Books.Google.com. CRC Press. Retrieved July 21, 2020.

ReferencesEdit