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]

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.