Let be a topological space, and let be an equivalence relation on . Write for the set of equivalence classes of under . The quotient topology on is the topology whose open sets are the subsets such that
is an open subset of . The space is called the quotient space of the space with respect to . It is often written .
The projection map which sends each element of to its equivalence class is always a continuous map. In fact, the map satisfies the stronger property that a subset of is open if and only if the subset of is open. In general, any surjective map that satisfies this stronger property is called a quotient map, and given such a quotient map, the space is always homeomorphic to the quotient space of under the equivalence relation
As a set, the construction of a quotient space collapses each of the equivalence classes of to a single point. The topology on the quotient space is then chosen to be the strongest topology such that the projection map is continuous.
For , one often writes for the quotient space obtained by identifying all the points of with each other.
|Date of creation||2013-03-22 12:39:40|
|Last modified on||2013-03-22 12:39:40|
|Last modified by||djao (24)|