Affine hull

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 induced by the quotient map.
• If $f$ is an inclusion map, then $f$ induces on $X_0$ a subspace topology.

References

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See also

• Natural topology