Quotient space (topology)

In topology, given a topological space X and an equivalence relation ~ on X, the quotient space, denoted X/~, is the set of equivalence classes of ~ equipped with the quotient topology.

The quotient topology on X/~ is defined as follows: Let q: X → X/~ be the quotient map that sends each element x in X to its equivalence class [x] in X/~. A subset U of X/~ is open in the quotient topology if and only if its preimage q^-1(U) is open in X. In other words, the quotient topology is the finest topology on X/~ that makes the quotient map q continuous.

The quotient topology is sometimes called the identification topology, and the process of forming the quotient space is called identifying points. The idea is that elements of X related by the equivalence relation are "glued together" to form a single point in the quotient space.

Properties:

• The quotient map q: X → X/~ is continuous by definition of the quotient topology.
• A function f: X/~ → Y to another topological space Y is continuous if and only if the composition f ∘ q: X → Y is continuous. This property makes the quotient topology "universal" in the sense that any continuous map from X that identifies equivalent points factors uniquely through a continuous map from the quotient space.
• If X is compact and the quotient map q is open, then X/~ is compact.
• If X is connected and the quotient map q is surjective, then X/~ is connected.

Related Concepts:

• Equivalence Relation: A binary relation that is reflexive, symmetric, and transitive.
• Quotient Map: The map that sends each element of a set to its equivalence class.
• Topological Space: A set equipped with a topology, which defines the open sets.
• Continuous Function: A function between topological spaces that preserves the open sets.