In general topology and related areas of mathematics, the initial topology (or induced topology[1][2] or strong topology or limit topology or projective topology) on a set with respect to a family of functions on is the coarsest topology on that makes those functions continuous.

The subspace topology and product topology 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 on that makes those functions continuous.

Definition

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Given a set   and an indexed family   of topological spaces with functions   the initial topology   on   is the coarsest topology on   such that each   is continuous.

Definition in terms of open sets

If   is a family of topologies   indexed by   then the least upper bound topology 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  [3]

If for every     denotes the topology on   then   is a topology on  , and the initial topology of the   by the mappings   is the least upper bound topology of the  -indexed family of topologies   (for  ).[3] Explicitly, the initial topology is the collection of open sets generated by all sets of the form   where   is an open set in   for some   under finite intersections and arbitrary unions.

Sets of the form   are often called cylinder sets. If   contains exactly one element, then all the open sets of the initial topology   are cylinder sets.

Examples

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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 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 forms of its dual space.
  • Given a family of topologies   on a fixed set   the initial topology on   with respect to the functions   is the supremum (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

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Characteristic property

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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  [4]

 
Characteristic property of the initial topology

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  [4]

Evaluation

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By the universal property of the product topology, we know that any family of continuous maps   determines a unique continuous map  

This map is known as the evaluation map.[citation needed]

A family of maps   is said to separate points in   if for all   in   there exists some   such that   The family   separates points if and only if the associated evaluation map   is injective.

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 if and only if these maps separate points on  [3]

Transitivity of the initial topology

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If   has the initial topology induced by the  -indexed family of mappings   and if for every   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  [5] 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  [4]

The product topology on   is equal to the initial topology induced by the canonical projections   as   ranges over  [4] Consequently, the initial topology on   induced by   is equal to the inverse image of the product topology on   by the evaluation map  [4] Furthermore, if the maps   separate points on   then the evaluation map is a homeomorphism onto the subspace   of the product space  [4]

Separating points from closed sets

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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 sets   in   and all   there exists some   such that   where   denotes the closure operator.

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

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If   is a family of uniform structures on   indexed by   then the least upper bound uniform structure 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  [6] 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  [6]

Now suppose that   is a family of maps and for every   let   be a uniform structure on   Then the initial uniform structure of the   by the mappings   is the unique coarsest uniform structure   on   making all   uniformly continuous.[6] It is equal to the least upper bound uniform structure of the  -indexed family of uniform structures   (for  ).[6] The topology on   induced by   is the coarsest topology on   such that every   is continuous.[6] The initial uniform structure   is also equal to the coarsest uniform structure such that the identity mappings   are uniformly continuous.[6]

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  [6] 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  [6] (or equivalently, if and only if the evaluation map   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  [6]

Cauchy filter: A filter   on   is a Cauchy filter on   if and only if   is a Cauchy prefilter on   for every  [6]

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

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In the language of category theory, the initial topology construction can be described as follows. Let   be the functor from a discrete category   to the category of topological spaces   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 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 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

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References

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  1. ^ Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
  2. ^ Adamson, Iain T. (1996). "Induced and Coinduced Topologies". A General Topology Workbook. Birkhäuser, Boston, MA. pp. 23–30. doi:10.1007/978-0-8176-8126-5_3. ISBN 978-0-8176-3844-3. Retrieved July 21, 2020. ... the topology induced on E by the family of mappings ...
  3. ^ a b c Grothendieck 1973, p. 1.
  4. ^ a b c d e f Grothendieck 1973, p. 2.
  5. ^ Grothendieck 1973, pp. 1–2.
  6. ^ a b c d e f g h i j Grothendieck 1973, p. 3.

Bibliography

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