A <basis of a topology> $\mathcal T$ is a collection $\mathcal B\subseteq\mathcal T$ such that every <open set> is a union of members of $\mathcal B$. Equivalently, for each open $U$ and every $x\in U$, there exists $B\in\mathcal B$ with $x\in B\subseteq U$. The empty union represents the empty set. In particular, the basis covers $X$, and any intersection of two basis elements is itself a union of basis elements.

Solution

Past exam of the options I course of the University of Oxford / 2013 / E / 1 / a / Solution

 Solution (source code)