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Can someone give me an example of a map which is not continuous such that $f(\{a\}) = f(\{b \})$ induces an equivalence relation $ \{ a \} \sim \{ b \} $?

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    On the space? Every map does.2012-12-07

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Let $f:\{a,b,c,\}\to\{0,1\}$. Let $f(a)=f(b)=0$ and $f(c)=1$. Let $x\sim y$ precisely if $f(x)=f(y)$. Then we have \begin{align} a & \sim a \\ a & \sim b \\ a & \not\sim c \\ \\ b & \sim a \\ b & \sim b \\ b & \not\sim c \\ \\ c & \not\sim a \\ c & \not\sim b \\ c & \sim c \end{align} This is an equivalence relation on the set $\{a,b,c\}$ with two equivalence classes: $\{a,b\}$ and $\{c\}$.