# 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

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.

A topology induced on by is the finest topology such that is continuous . This is a particular case of the final topology on .

A 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 induced by the quotient map.
- If is an inclusion map, then induces on a subspace topology.

## References

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