# 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 (*Y*_{i})_{i∈I} 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 *i* ∈ *I*, 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

- The quotient topology is the topology coinduced by the quotient map.
- The product topology is the topology induced by the projections .
- If is an inclusion map, then induces on the subspace topology.
- The weak topology is that induced by the dual on a topological vector space.
^{[1]}

## ReferencesEdit

- ^
^{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. - ^
^{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 ...

**^**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

- Natural topology
- The initial topology and final topology are used synonymously, though usually only in the case where the (co)inducing collection consists of more than one function.

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