is a type of topology (𝜏) on a set 𝑋
a set 𝑋 endowed with a quotient topology is called a quotient space
If ∼ is an equivalence relation on a topological space 𝑋, then the quotient space 𝑋/~ = {[𝑥] : 𝑥∈𝑋} and the quotient topology on 𝑋/∼ is defined to be:
- 𝜏_𝑋/∼ = { 𝐴⊆𝑋/~ | 𝑞⁻¹(𝐴) is open in 𝑋 }
where:
- 𝑞: 𝑋 → 𝑋/∼ defined by 𝑝 ↦ [𝑝] is the quotient map
- [𝑥] is the equivalence class (or in this case the coset) of 𝑥

Theorems

Examples

The following are quotient topologies

[0,1] / (0~1)
ℝ/~ where 𝑥~𝑦 if and only if 𝑥-𝑦 ∊ ℤ. This is often written as ℝ/ℤ. This is in fact homeomorphic to [0,1] / (0~1) and can be thought of as a circle.
ℝ²/ℤ², defined similarly to the above. By identifying the top and bottom of the unit square and the two sides, we see that this can be thought of as a torus
Let 𝐷² = { 𝑥∊ℝ² : |𝑥|≤1 } and ~ be generated by 𝑥~𝑦 if and only if |𝑥| = |𝑦| = 1. Then 𝐷²/𝑆² := 𝐷²/~ ≅ 𝑆²
𝑆¹/(𝑥 ~ -𝑥) ≅ 𝑆¹
𝑆²/(𝑥 ~ -𝑥) is known as a cross-cap