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.

See also

• Natural topology