Induced topology

In topology and related areas of mathematics, an induced topology on a topological space is a topology which is "optimal" for some function from/to this topological space.

Definition

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

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

If $\tau _{1}$ is a topology on $X_{1}$ , then a 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 $\tau _{1}=\{f(U_{0})|U_{0}\in \tau _{0}\}$ 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.

A topology $\tau _{1}$ induced 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 $X_{1}$ .

A 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 $X_{0}$ .

Examples

The quotient topology is the topology coinduced by the quotient map.
If $f$ is an inclusion map, then $f$ induces on $X_{0}$ a subspace topology.

References

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

Induced topology

Definition

Examples

References

See also