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Section Summary
Important ideas that we discussed in this section include the following.
Let \((X,\tau_X)\) 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
\begin{equation*}
\{U \subseteq Y \mid p^{-1}(U) \in \tau_X\}\text{.}
\end{equation*}
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.
