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added within_continuous_patch #1883

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ca0ca50
added within_continuous_patch
holgerthies Mar 5, 2026
083a516
changelog
holgerthies Mar 5, 2026
a11f449
linting
affeldt-aist May 1, 2026
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3 changes: 3 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -79,6 +79,9 @@
- in `measure_extension.v`:
+ definition `caratheodory_measure`

- in `subspace_topology.v`:
+ lemma `withinU_continuous_patch`

### Changed

- moved from `measurable_structure.v` to `classical_sets.v`:
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22 changes: 22 additions & 0 deletions theories/topology_theory/subspace_topology.v
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Expand Up @@ -711,3 +711,25 @@ HB.instance Definition _ :=
@isContinuous.Build (subspace A) Z (g \o f) continuous_comp_subproof.

End continuous_fun_comp.

Section continuous_patch.
Context {U V : topologicalType}.
Variables (A B : set U) (f g : U -> V).
Hypothesis contf : {within A, continuous f}.
Hypothesis contg : {within B, continuous g}.
Hypothesis closedA : closed A.
Hypothesis closedB : closed B.
Hypothesis AB_fg : forall x, x \in A `&` B -> f x = g x.

Lemma withinU_continuous_patch : {within A `|` B, continuous (patch g A f)}.
Proof.
pose gAf := patch g A f.
apply: withinU_continuous => //.
- suff : {in A, f =1 gAf} by move/subspace_eq_continuous; exact.
by rewrite /gAf /patch => r ->.
- suff : {in B, g =1 gAf} by move/subspace_eq_continuous; exact.
move=> r rB; rewrite /gAf /patch; case: ifPn => // rA.
by apply/esym/AB_fg; rewrite in_setI rA.
Qed.

End continuous_patch.
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