# A question about a basis for a topology

If a subset $B$ of a powerset $P(X)$ has the property that finite intersections of elements of $B$ are empty or again elements of $B$, does the collection of all unions of sets from $B$ form a topology on $X$ then?

My book A Taste of Topology says this is indeed the case, but I wonder how you can guarantee that $X$ will be open. For example, the empty set has the required properties, so that would mean that the empty set is a topology for any set $X$, which is impossible.

By a standard convention $X$ is vacuously equal to $\bigcap\varnothing$. To see why this should be so, ask yourself how you would demonstrate that some $x\in X$ is not in some intersection $\bigcap\mathscr{A}$ of subsets of $X$: you’d show that there was some $A\in\mathscr{A}$ such that $x\notin A$. If $\mathscr{A}=\varnothing$, you can’t even find an $A\in\mathscr{A}$, let alone one that doesn’t contain $x$!