Q1: Let $X$ be a set and $\sim$ be an equivalence relation on $X$. What is not a property of $\sim$?

A1: $x \sim x$ for all $x \in X$

A2: If $x \sim y$, then also $y \sim x$

A3: If $x \sim y$ or $x \sim z$, then also $y \sim z$.

A4: If $x \sim y$ and $y \sim z$, then also $x \sim z$.

Q2: Let $(X, \mathcal{T}) $ be a topological space and $\sim$ be an equivalence relation on $X$. What is the definition of the canonical projection $q$?

A1: $q(x) = [x]_{\sim}$

A2: $q([x]_{\sim}) = x$

A3: $q(x) = X/_{\sim}$

Q3: Let $(X, \mathcal{T}) $ be a topological space and $\sim$ be an equivalence relation on $X$. What is the definition of the quotient topology on $X/_{\sim}$?

A1: A set $U$ in $X/_{\sim}$ is open if $q[U]$ is open in $X$.

A2: A set $U$ in $X/_{\sim}$ is open if $q^{-1}[U]$ is open in $X$.

A3: A set $U$ in $X/_{\sim}$ is open if $q[U]$ is not open in $X$.