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]

For the topology on a subspace of a topological space, see Subspace topology.

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 ${\displaystyle X_{0},X_{1}}$ be sets, ${\displaystyle f:X_{0}\to X_{1}}$.

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

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

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

There are equivalent definitions below.

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

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

General case

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

${\displaystyle f_{i}:X\to Y_{i},}$

the topology ${\displaystyle \tau }$ on ${\displaystyle X}$ induced by these functions is the coarsest topology on X such that each

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

is continuous.[1][2]

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

Examples

• The quotient topology is the topology coinduced by the quotient map.
• If ${\displaystyle f:X_{0}\to X}$ is an inclusion map, then ${\displaystyle f}$ induces on ${\displaystyle X_{0}}$ the subspace topology.
• The weak topology is that induced by the dual on a topological vector space.[1]

References

1. Rudin, Walter (January 1, 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.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
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 ...
3. 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.