Past exam of the mathematics course of the University of Cambridge / 2008 / ib / Paper 1 / 12F / Solution

A topology on $X$ is a family $\mathcal{T}$ of subsets containing $\varnothing$ and $X$, closed under arbitrary unions and finite intersections. Its members are the open sets. A Hausdorff space separates each pair of distinct points by disjoint open neighbourhoods; a compact space is one for which every open cover has a finite subcover.