# Is every element from the finite complement topology on $\mathbb R$ generated by this subbasis a basis element?

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?

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.