Final topology

Summary

In general topology and related areas of mathematics, the final topology[1] (or coinduced,[2] strong, colimit, or inductive[3] 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 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

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Given a set   and an  -indexed family of topological spaces   with associated functions   the final topology on   induced by the family of functions   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  .

The family   of functions that induces the final topology on   is usually a set of functions. But the same construction can be performed if   is a proper class of functions, and the result is still well-defined in Zermelo–Fraenkel set theory. In that case there is always a subfamily   of   with   a set, such that the final topologies on   induced by   and by   coincide. For more on this, see for example the discussion here.[4] As an example, a commonly used variant of the notion of compactly generated space is defined as the final topology with respect to a proper class of functions.[5]

Examples

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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. 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  

Given a topological space   and a family   of subsets of   each having the subspace topology, the final topology   induced by all the inclusion maps of the   into   is finer than (or equal to) the original topology   on   The space   is called coherent with the family   of subspaces if the final topology   coincides with the original topology   In that case, a subset   will be open in   exactly when the intersection   is open in   for each   (See the coherent topology article for more details on this notion and more examples.) As a particular case, one of the notions of compactly generated space can be characterized as a certain coherent topology.

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

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Characterization via continuous maps

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Given functions   from topological spaces   to the set  , the final topology on   with respect to these functions   satisfies 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

This property characterizes the final topology in the sense that if a topology on   satisfies the property above for all spaces   and all functions  , then the topology on   is the final topology with respect to the  

Behavior under composition

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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

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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   and   is continuous for every index   then   must be strictly coarser than   (meaning that   and   this will be written  ) and moreover, for any subset   the topology   will also be strictly coarser than the final topology   that   induces on   (because  ); that is,  

Suppose that in addition,   is an  -indexed 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).

Final topology on the direct limit of finite-dimensional Euclidean spaces

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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 (respectively, closed) in   if and only if for every   the set   is an open (respectively, closed) subset of   The topology   is coherent with the 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

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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

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Notes

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  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

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  1. ^ Bourbaki, Nicolas (1989). General topology. Berlin: Springer-Verlag. p. 32. ISBN 978-3-540-64241-1.
  2. ^ Singh, Tej Bahadur (May 5, 2013). Elements of Topology. CRC Press. ISBN 9781482215663. Retrieved July 21, 2020.
  3. ^ Császár, Ákos (1978). General topology. Bristol [England]: A. Hilger. p. 317. ISBN 0-85274-275-4.
  4. ^ "Set theoretic issues in the definition of k-space or final topology wrt a proper class of functions". Mathematics Stack Exchange.
  5. ^ Brown 2006, Section 5.9, p. 182.

References

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  • Brown, Ronald (June 2006). Topology and Groupoids. North Charleston: CreateSpace. ISBN 1-4196-2722-8.
  • Willard, Stephen (1970). General Topology. Addison-Wesley Series in Mathematics. Reading, MA: Addison-Wesley. ISBN 9780201087079. Zbl 0205.26601.. (Provides a short, general introduction in section 9 and Exercise 9H)
  • Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.