Past exam of the options I course of the University of Oxford / 2010 / E / E1 / a

A topology 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 is closed precisely when its complement in $X$ is open.

The subspace topology on $Y$ isIf $Y$ is open in $X$, every such intersection is open in $X$. Conversely, every ambient open set contained in $Y$ equals its intersection with $Y$. Thus the relatively open sets are exactly the ambient open subsets contained in $Y$.

The closure of $A$ is the smallest closed subset of $X$ containing it:Equivalently, a point belongs to $\overline{A}$ exactly when every open neighborhood of it meets $A$.