factor space

The quotient space X/~ of a topological space X and an equivalence relation ~ on X is the set of equivalence classes of points in X (under the equivalence relation ~) together with the following topology given to subsets of X/~: a subset U of X/~ is called open iff union _([a] element U) a is open in X. Quotient spaces are also called factor spaces. This can be stated in terms of maps as follows: if q:X->X/~ denotes the map that sends each point to its equivalence class in X/~, the topology on X/~ can be specified by prescribing that a subset of X/~ is open iff q^(-1)[the set] is open.

equivalence relation | Lie group quotient space | topological space