Check the details of Example 3 of §22: for the map \(p: \mathbb{R} \to A = \ts{a, b, c}\) defined by
\[ p(x) = \begin{cases} a & \text{if } x > 0, \\ b & \text{if } x < 0, \\ c & \text{if } x = 0, \end{cases} \]
determine the quotient topology on \(A\) induced by \(p\).
Solution
1. The preimages of the subsets of \(A\) are \(p^{-1}(\varnothing)=\varnothing\), \(p^{-1}(\{a\})=(0,\infty)\), \(p^{-1}(\{b\})=(-\infty,0)\), \(p^{-1}(\{a,b\})=\mathbb R-\{0\}\), \(p^{-1}(A)=\mathbb R\), which are open, and \(p^{-1}(\{c\})=\{0\}\), \(p^{-1}(\{a,c\})=[0,\infty)\), \(p^{-1}(\{b,c\})=(-\infty,0]\), which are not open.
The preimages follow from the definition of \(p\). The last three each contain \(0\) but no open interval about \(0\).
2. Q.E.D.
A subset \(U\subseteq A\) is open in the quotient topology if and only if \(p^{-1}(U)\) is open, so by step 1 the quotient topology is \[ \boxed{\bigl\{\varnothing,\{a\},\{b\},\{a,b\},A\bigr\}}. \]