# 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

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.

## References

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