Induced topology

In topology and related areas of mathematics, an induced topology on a topological space is a topology that makes a given (inducing) function or collection of functions continuous from this topological space.[1][2]

A coinduced topology or final topology makes the given (coinducing) collection of functions continuous to this topological space.[3]

DefinitionEdit

The case of just one functionEdit

Let   be sets,  .

If   is a topology on  , then the topology coinduced on   by   is  .

If   is a topology on  , then the topology induced on   by   is  .

The easy way to remember the definitions above is to notice that finding an inverse image is used in both. This is because inverse image preserves union and intersection. Finding a direct image does not preserve intersection in general. Here is an example where this becomes a hurdle. Consider a set   with a topology  , a set   and a function   such that  . A set of subsets   is not a topology, because   but  .

There are equivalent definitions below.

The topology   coinduced on   by   is the finest topology such that   is continuous  . This is a particular case of the final topology on  .

The topology   induced on   by   is the coarsest topology such that   is continuous  . This is a particular case of the initial topology on  .

General caseEdit

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

 

the topology   on   induced by these functions is the coarsest topology on X such that each

 

is continuous.[1][2]

Explicitly, the induced 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

ReferencesEdit

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

SourcesEdit

  • Hu, Sze-Tsen (1969). Elements of general topology. Holden-Day.

See alsoEdit