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A subbasis of a topology $(X,\tau)$ is defined as a collection of sets such that the union of such sets is $X$, and the union of finite intersections of subbase sets form open sets.

My question is: are these finite intersections of subbase sets also open sets of the topology?

Thanks in advance!

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Yes. Note that if $A$ is an element of the subbase, then $A$ is the intersection of a single element ($A$ itself!), which is a finite intersection of subbase elements. It is also the union of a single finite intersection.

Therefore the elements of the subbase are open, so their finite intersections are open as well.

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Certainly. Consider a union comprising of a single set $S$ which is a finite intersection of certain elements from the sub-base. Then, by the given definition, this union ($S$) must be open. Thus, every finite intersection of sub-base sets is open.

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