Require Export TopologicalSpaces NeighborhoodBases.
From ZornsLemma Require Export CountableTypes.
From Coq Require ProofIrrelevance ClassicalChoice.
Global Set Asymmetric Patterns.
Definition first_countable (X:TopologicalSpace) : Prop :=
forall x:point_set X, exists NBx:Family (point_set X),
neighborhood_basis NBx x /\ Countable NBx.
Require Export Nets.

Lemma first_countable_sequence_closure:
forall (X:TopologicalSpace) (S:Ensemble (point_set X)) (x:point_set X),
first_countable X -> In (closure S) x ->
exists y:Net nat_DS X, (forall n:nat, In S (y n)) /\
net_limit y x.