Induced topology
In topology and related areas of mathematics, an induced topology on a topological space is a topology that makes the inducing function continuous from/to this topological space.
Definition
Let be sets, .
If is a topology on , then the topology coinduced on by is .
If is a topology on , then the topology induced on by is .
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 with a topology , a set and a function such that . A set of subsets is not a topology, because but .
There are equivalent definitions below.
The topology coinduced on by is the finest topology such that is continuous . This is a particular case of the final topology on .
The topology induced on by is the coarsest topology such that is continuous . This is a particular case of the initial topology on .
Examples
- The quotient topology is the topology coinduced by the quotient map.
- If is an inclusion map, then induces on the subspace topology.
References
- Hu, Sze-Tsen (1969). Elements of general topology. Holden-Day.
See also
- Natural topology
- The initial topology and final topology are generalizations of this concept.