Past exam of the options I course of the University of Oxford / 2010 / E / E2 / a

A space is Hausdorff if any two distinct points have disjoint open neighborhoods: for $x\ne y$, there are open $U\ni x$ and $V\ni y$ with $U\cap V=\varnothing$.

The map $f:X\to Y$ is continuous ifThis condition concerns inverse images of open sets; images of open sets need not be open.

Let $p:X\to X/R$ send a point to its equivalence class. The quotient topology isIt is the finest topology on the quotient making $p$ continuous.