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.
The case of just one function
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 on induced by these functions is the coarsest topology on X such that each
Explicitly, the induced topology is the collection of open sets generated by all sets of the form , where is an open set in for some i ∈ I, under finite intersections and arbitrary unions. The sets are often called cylinder sets. If I contains exactly one element, all the open sets of are cylinder sets.
- The quotient topology is the topology coinduced by the quotient map.
- The product topology is the topology induced by the projections .
- If is an inclusion map, then induces on the subspace topology.
- The weak topology is that induced by the dual on a topological vector space.
- Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
- Adamson, Iain T. (1996). "Induced and Coinduced Topologies". A General Topology Workbook. Birkhäuser, Boston, MA. p. 23. doi:10.1007/978-0-8176-8126-5_3. Retrieved July 21, 2020.
... the topology induced on E by the family of mappings ...
- Singh, Tej Bahadur (May 5, 2013). "Elements of Topology". Books.Google.com. CRC Press. Retrieved July 21, 2020.
- Hu, Sze-Tsen (1969). Elements of general topology. Holden-Day.