Let be sets, .
If is a topology on , then a topology coinduced on by is .
If is a topology on , then a 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 quotient topology is the topology induced by the quotient map.
- If is an inclusion map, then induces on a subspace topology.
- Hu, Sze-Tsen (1969). Elements of general topology. Holden-Day.
|This topology-related article is a stub. You can help Wikipedia by expanding it.|