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feat: a topology with a basis of clopens is completely regular #37317

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feat: a topology with a basis of clopens is completely regular #37317

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dcd5047
prove the theorem
vihdzp Mar 28, 2026
6f4e35c
Update Mathlib/Topology/Separation/CompletelyRegular.lean
vihdzp Mar 29, 2026
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11 changes: 11 additions & 0 deletions Mathlib/Topology/Separation/CompletelyRegular.lean
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Expand Up @@ -11,6 +11,8 @@ public import Mathlib.Topology.Compactification.StoneCech
public import Mathlib.Topology.Order.Lattice
public import Mathlib.Analysis.Real.Cardinality

import Mathlib.Topology.Algebra.Indicator

/-!
# Completely regular topological spaces.

Expand Down Expand Up @@ -193,6 +195,15 @@ lemma completelyRegularSpace_iff_isInducing_stoneCechUnit :
mp _ := isInducing_stoneCechUnit
mpr hs := hs.completelyRegularSpace

theorem CompletelyRegularSpace.of_isTopologicalBasis_clopens
(h : TopologicalSpace.IsTopologicalBasis {s : Set X | IsClopen s}) :
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CompletelyRegularSpace X where
completely_regular x K hK hx := by
obtain ⟨s, hs, hx, hsK⟩ := h.exists_subset_of_mem_open hx hK.isOpen_compl
refine ⟨sᶜ.indicator 1, ?_, by simpa, fun x hx ↦ indicator_of_mem ?_ _⟩
· exact hs.compl.continuous_indicator continuous_const
· exact (mem_compl_iff s x).mpr fun hs ↦ hsK hs hx

open TopologicalSpace Cardinal in
theorem CompletelyRegularSpace.isTopologicalBasis_clopens_of_cardinalMk_lt_continuum
[CompletelyRegularSpace X] (hX : Cardinal.mk X < continuum) :
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