proposition: Lemma continuous_open_basis: forall (B:Family Y), open_basis B -> (forall V:Ensemble Y, In B V -> open (inverse_image f V)) -> continuous.
proof: Proof. intros. apply pointwise_continuity. intro. pose proof (open_basis_to_open_neighborhood_basis B (f x) H). apply open_neighborhood_basis_is_neighborhood_basis in H1. apply (continuous_at_neighborhood_basis _ _ H1). intros. destruct H2 as [[? ?]]. apply open_neighborhood_is_neighborhood. split; try constructor; auto. Qed.
imports: TopologicalSpaces Neighborhoods OpenBases NeighborhoodBases Subbases