Describe how to make a topology from a subbasis, and what (if any) requirements there are on the subbasis.
Solution
Given a collection \(\mathcal S\subseteq\mathcal P(X)\), the topology generated by \(\mathcal S\) is the smallest topology on \(X\) containing every member of \(\mathcal S\). Equivalently, let \(\mathcal B\) consist of all finite intersections of members of \(\mathcal S\), including the empty intersection \(X\). Then arbitrary unions of members of \(\mathcal B\) form the generated topology.
Thus, with the empty intersection \(X\) allowed, there is no condition on \(\mathcal S\): every collection of subsets generates a topology.
Under the common convention that a subbasis must itself cover \(X\) and that one takes only nonempty finite intersections, the requirement is \[ \bigcup_{S\in\mathcal S}S=X. \] This guarantees that the finite intersections cover \(X\) and form a basis. The two conventions generate the same notion after adjoining \(X\) to \(\mathcal S\) when necessary.