Past exam of the probability, statistics and options I course of the University of Oxford / 2009 / E / 1 / a

The closed sets contain the empty set and the whole space, are closed under arbitrary intersections, and are closed under finite unions. These conditions are equivalent, by complements, to the open-set axioms of a topology.

A map is continuous when the inverse image of every open set is open. Equivalently, the inverse image of every closed set is closed.

The definition of convergence of a sequence in a topological space to $\ell$ is that every open neighborhood $U$ of $\ell$ contains all sufficiently late terms: there exists $N$ such that $x_n\in U$ for every $n\ge N$.