Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / E / 1 / a

A topology $\mathcal{T}$ on $X$ contains $\varnothing$ and $X$, is closed under arbitrary unions, and is closed under finite intersections. Its members are the open sets. A subset $F\subseteq X$ is closed precisely when $X\setminus F$ is open.

The subspace topology on $Y$ isIf $Y$ is open in $X$, each such intersection is open in $X$. Conversely, an open subset of $X$ contained in $Y$ is its own intersection with $Y$. Thus the open sets of $Y$ are exactly the open subsets of $X$ contained in $Y$ in this case.

The closure $\overline{A}$ is the intersection of all closed subsets of $X$ containing $A$. It is the smallest such closed subset. Equivalently, $x\in \overline{A}$ exactly when every open neighborhood of $x$ meets $A$.