ART

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]

Definition
The case of just one function

Let \( X_{0},X_{1} \) be sets, \( f:X_{0}\to X_{1}. \)

If \( \tau _{0} \)is a topology on \( X_{0} \), then the topology coinduced on \( X_{1} \) by f is \( (U_{1})\in \tau _{0}\}} \{U_{1}\subseteq X_{1}|f^{-1}(U_{1})\in \tau _{0}\}. \)

If \( \tau _{1} \) is a topology on \( X_{1} \) , then the topology induced on \( X_{0} \) by f is \( \{f^{-1}(U_{1})|U_{1}\in \tau _{1}\}. \)

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 \( X_{0}=\{-2,-1,1,2\} \) with a topology \( \{\{-2,-1\},\{1,2\}\} \), a set \( X_{1}=\{-1,0,1\} \) and a function \( f:X_{0}\to X_{1} \) such that \( f(-2)=-1,f(-1)=0,f(1)=0,f(2)=1 \) . A set of subsets \( {\displaystyle \tau _{1}=\{f(U_{0})|U_{0}\in \tau _{0}\}} \tau _{1}=\{f(U_{0})|U_{0}\in \tau _{0}\} \) is not a topology, because \( \{\{-1,0\},\{0,1\}\}\subseteq \tau _{1} \) but \( \{-1,0\}\cap \{0,1\}\notin \tau _{1}. \)

There are equivalent definitions below.

The topology \( \tau _{1} \) coinduced on\( X_{1} \) by f is the finest topology such that f is continuous \( (X_{0},\tau _{0})\to (X_{1},\tau _{1}) \). This is a particular case of the final topology on \( X_{1} \).

The topology \( \tau _{0} \) induced on \( X_{0} \) by f is the coarsest topology such that f is continuous \( (X_{0},\tau _{0})\to (X_{1},\tau _{1}) \). This is a particular case of the initial topology on \( X_{0} \).

General case

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

\( {\displaystyle f_{i}:X\to Y_{i},} \)

the topology \( \tau \) on X induced by these functions is the coarsest topology on X such that each

\( f_{i}:(X,\tau )\to Y_{i} \)

is continuous.[1][2]

Explicitly, the induced topology is the collection of open sets generated by all sets of the form \( f_{i}^{{-1}}(U) \), where U is an open set in \( Y_{i} \) for some i ∈ I, under finite intersections and arbitrary unions. The sets \( f_{i}^{{-1}}(U) \) are often called cylinder sets. If I contains exactly one element, all the open sets of \( (X,\tau ) \) are cylinder sets.

Examples

The quotient topology is the topology coinduced by the quotient map.
The product topology is the topology induced by the projections \( {\displaystyle {\text{proj}}_{j}:X\to X_{j}}. \)
If \( {\displaystyle f:X_{0}\to X} \) is an inclusion map, then f induces on \( X_{0} \) the subspace topology.
The weak topology is that induced by the dual on a topological vector space.[1]

References

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.
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.

Sources

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

See also

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.

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

World

Index

Hellenica World - Scientific Library

Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License