Induced topology

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 ${\displaystyle X_{0}}$.

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.