= Solution
For a sequence $\epsilon$, let $C_n(\epsilon)$ be its prefix <cylinder set>, consisting of sequences agreeing with $\epsilon$ in the first $n$ coordinates. The specified collection consists exactly of unions of such <cylinder sets>.
The empty set meets the defining condition vacuously, and $X$ meets it because every <cylinder set> is contained in $X$. For a union $\bigcup_\lambda Y_\lambda$, a point lies in one $Y_\lambda$ and has a prefix <cylinder set> contained in that set, hence contained in the union. For a finite intersection, a point has a cylinder neighbourhood in every constituent. Taking the largest of the finitely many prefix lengths gives one <cylinder set> contained in all of them. Therefore the collection contains the empty and whole sets, is closed under arbitrary unions and finite intersections, and \b[is a <topology> on $X$], the prefix topology of <Cantor space>.
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