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A subbasis $\mathbb S$ for the finite complement topology is the set of all $\mathbb R-x$ where $x \in \mathbb R$. From Munkres I know that the collection of all finite intersections of elements from $\mathbb S$ forms a basis.

Any finite intersection of elements from $\mathbb S$ will be a basis element. That is, any set missing finitely many points will be a basis element. But that is every open set in the finite complement topology, besides $\mathbb R$.

So, is every element from the finite complement topology generated by this subbasis a basis element?

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Yes, your reasoning is correct. The basis is the entire topology. Note that even the whole space is in the basis, since it is the empty intersection. The empty set maybe has to be added manually though.

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No, since the constructed basis will not contain $\varnothing$ which is an element of the topology.

The constructed basis will contain every other element of the topology, so including $\mathbb R$ which is the empty intersection of subbase elements.

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