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unif cont -> cont #2113

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3 changes: 3 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Original file line number Diff line number Diff line change
Expand Up @@ -4,6 +4,9 @@

### Added

- in `uniform_structure.v`:
+ lemma `unif_continuous_continuous`

### Changed

### Renamed
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22 changes: 22 additions & 0 deletions theories/topology_theory/uniform_structure.v
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Expand Up @@ -328,6 +328,28 @@ Qed.
Definition unif_continuous (U V : uniformType) (f : U -> V) :=
(fun xy => (f xy.1, f xy.2)) @ entourage --> entourage.

Lemma unif_continuous_continuous (U V : uniformType) (f : U -> V) :
unif_continuous f -> continuous f.
Proof.
move=> ucf /= u N fuN.
have [V0 entV0 V0N] : exists2 V0 : set (V * V),
entourage V0 & xsection V0 (f u) `<=` N.
move: fuN; rewrite -filter_from_entourageE => -[V1 entV1 V1fuN].
by exists V1.
have [U0 entU0 U0V0] : exists2 U0 : set (U * U),
entourage U0 & (fun x => (f x.1, f x.2)) @` U0 `<=` V0.
exists ((fun x => (f x.1, f x.2)) @^-1` V0).
exact: ucf.
by move=> [_ _]/= [x V0fx [<- <-]].
have fU0uN : f @` (xsection U0 u) `<=` N.
move=> _/= [u0 U0u0 <-]; apply: V0N => /=.
by apply/mem_set/U0V0 => /=; exists (u, u0) => //=; exact/set_mem.
apply/nbhsP; exists U0 => //.
have := @preimage_subset _ _ f _ _ fU0uN.
apply: subset_trans => u0/= U0uu0.
by exists u0.
Qed.

Definition entourage_set (U : uniformType) (A : set ((set U) * (set U))) :=
exists2 B, entourage B & forall PQ, A PQ -> forall p q,
PQ.1 p -> PQ.2 q -> B (p,q).
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