Summary

Section Summary

Important ideas that we discussed in this section include the following.

🔗

Let $(X, τ_{X})$ be a topological space, let $Y$ be a set, and let $p : X \to Y$ be a surjection. The quotient topology on $Y$ is the set

${U \subseteq Y ∣ p^{- 1} (U) \in τ_{X}} .$
🔗
The function $p$ is a quotient topology as in the previous bullet is called a quotient map and the space $Y$ is a quotient space.
🔗
A circle, a Möbius strip, a torus, and a sphere can all be realized as quotient spaces.