Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / E / 2 / a

A Hausdorff space is one in which any distinct points $x,y$ have disjoint open neighborhoods: there exist open sets $U,V$ with $x\in U$, $y\in V$ and $U\cap V=\varnothing$.

A map $f:X\to Y$ is continuous if the inverse image of every open subset of $Y$ is open in $X$:

For the natural projection $p:X\to X/R$, the quotient topology declares $U\subseteq X/R$ open exactly when $p^{-1}(U)$ is open in $X$. It is the finest topology on the quotient making $p$ continuous.