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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.

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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 $x$!

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It is usually taken that the empty intersection of subsets of a set is the entire set, similar to how the empty product is often taken to be the multiplicative identity or the empty sum is often taken to be the additive identity.

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