Final topology

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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.

Contents

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 inclusion maps. 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.

Definition

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 .

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 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.

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
Characteristic property of the final topology FinalTopology-01.png
Characteristic property of the final topology

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 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 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 finer 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 strictly coarser then (in symbols, which means and ) 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 subspaces

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 spaces

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 inclusion map defined by so that its 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 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 inclusion map 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 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 Top that selects the spaces for Let be the diagonal functor from Top to the functor category TopJ (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 SetJ 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 left adjoint to the corresponding forgetful functor.

See also

Notes

  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

Citations

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

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