Past exam of the options I course of the University of Oxford / 2009 / E / E1 / a

The family $\mathcal{F}$ of closed sets contains $\varnothing$ and $X$, is closed under arbitrary intersection, and is closed under finite union. Conversely these axioms define a topology by declaring the complements of the members of $\mathcal{F}$ to be the open sets.

The map is continuous when the preimage $f^{-1}(U)$ is open in $X$ for every open set $U$ in $Y$. Equivalently the preimage of every closed set in $Y$ is closed in $X$, since preimages commute with complementation.

Convergence of a sequence to $\ell$ means that every open neighbourhood $U$ of $\ell$ contains all sufficiently late terms: there is $N$ such that $x_n\in U$ for all $n\ge N$. The threshold may depend on $U$.