• Intuitively speaking, the points of each equivalence class are identified or “glued together” for forming a new topological space. For example, identifying the points of a sphere that belong to the same diameter produces the projective plane as a quotient space
• gluing - topologists talk of gluing points together. If 𝑋 is a topological space, gluing the points 𝑥 and 𝑦 in 𝑋 means considering the quotient space obtained from the equivalence relation 𝑎∼𝑏 if and only if either:
  • 𝑎=𝑏
  • 𝑎=𝑥,𝑏=𝑦
  • 𝑎=𝑦,𝑏=𝑥

Illustration of the construction of a topological sphere as the quotient space of a disk, by gluing together to a single point the points (in blue) of the boundary of the disk.