Initial topology

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

DefinitionEdit

Given a set X and an indexed family (Yi)iI of topological spaces with functions

 

the initial topology   on   is the coarsest topology on X such that each

 

is continuous.

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 iI, under finite intersections and arbitrary unions. The sets   are often called cylinder sets. If I contains exactly one element, all the open sets of   are cylinder sets.

ExamplesEdit

Several topological constructions can be regarded as special cases of the initial topology.

PropertiesEdit

Characteristic propertyEdit

The initial topology on X can be characterized by the following characteristic property:
A function   from some space   to   is continuous if and only if   is continuous for each i ∈ I.

Note that, despite looking quite similar, this is not a universal property. A categorical description is given below.

EvaluationEdit

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.

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

The evaluation map f will be a topological embedding if and only if X has the initial topology determined by the maps   and this family of maps separates points in X.

Separating points from closed setsEdit

If a space X comes equipped with a topology, it is often useful to know whether or not the topology on X is the initial topology induced by some family of maps on X. This section gives a sufficient (but not necessary) condition.

A family of maps {fi: XYi} separates points from closed sets in X if for all closed sets A in X and all x not in A, there exists some i such that

 

where cl denotes the closure operator.

Theorem. A family of continuous maps {fi: XYi} separates points from closed sets if and only if the cylinder sets  , for U open in Yi, form a base for the topology on X.

It follows that whenever {fi} separates points from closed sets, the space X has the initial topology induced by the maps {fi}. The converse fails, since generally the cylinder sets will only form a subbase (and not a base) for the initial topology.

If the space X is a T0 space, then any collection of maps {fi} that separates points from closed sets in X must also separate points. In this case, the evaluation map will be an embedding.

Categorical descriptionEdit

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  , i.e.: 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 alsoEdit

ReferencesEdit

  1. ^ Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. 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. p. 23. doi:10.1007/978-0-8176-8126-5_3. Retrieved July 21, 2020. ... the topology induced on E by the family of mappings ...

SourcesEdit