Induced topology

In topology and related areas of mathematics, an induced topology on a topological space is a topology that makes a given (inducing) function or collection of functions continuous from this topological space.

A coinduced topology or final topology makes the given (coinducing) collection of functions continuous to this topological space.

Definition

The case of just one function

Let $X_0,X_1$ be sets, $f:X_0\to X_1$.

If ${\tau }_0$ is a topology on $X_0$, then the 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 the 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.

The topology ${\tau }_1$ coinduced 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 $X_1$.

The 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 $X_0$.

General case

Given a set X and an indexed family (Y_i)_i∈I of topological spaces with functions

$f_i:X\to Y_i,$

the topology $\tau$ on $X$ induced by these functions is the coarsest topology on X such that each

$f_i:(X,\tau )\to Y_i$

is continuous.

Explicitly, the induced topology is the collection of open sets generated by all sets of the form $f_i^{-1}(U)$, where $U$ is an open set in $Y_i$ for some i ∈ I, under finite intersections and arbitrary unions. The sets $f_i^{-1}(U)$ are often called cylinder sets. If I contains exactly one element, all the open sets of $(X,\tau )$ are cylinder sets.

Examples

• The quotient topology is the topology coinduced by the quotient map.
• The product topology is the topology induced by the projections ${\text{proj}}_j:X\to X_j$.
• If $f:X_0\to X$ is an inclusion map, then $f$ induces on $X_0$ the subspace topology.
• The weak topology is that induced by the dual on a topological vector space.

See also

• Natural topology
• The initial topology and final topology are used synonymously, though usually only in the case where the (co)inducing collection consists of more than one function.

Sources

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