# 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 (Yi)iI 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

$f_{i}:(X,\tau )\to Y_{i}$

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 iI, 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.