End Subbasis.
Arguments subbasis {X}.
Section build_from_subbasis.
Variable X:Type.
Variable S:Family X.
Definition Build_TopologicalSpace_from_subbasis : TopologicalSpace.
refine (Build_TopologicalSpace_from_open_basis
(finite_intersections S) _ _).
red; intros.
exists (Intersection U V); repeat split; trivial.
apply intro_intersection; trivial.
destruct H1; assumption.
destruct H1; assumption.
destruct H2; assumption.
destruct H2; assumption.
red; intro.
exists Full_set.
split; constructor.
Defined.

Lemma Build_TopologicalSpace_from_subbasis_subbasis:
@subbasis Build_TopologicalSpace_from_subbasis S.