feat(Auxiliary): hereditary separability of linearly ordered topological spaces - #499
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Add `BrownianMotion/Auxiliary/SeparableSpace.lean`: a separable linearly ordered topological space is hereditarily separable (`Set.separableSpace`), with the supporting order-convex neighbourhood basis (`exists_isOpen_ordConnected_mem_subset`, `isTopologicalBasis_isOpen_ordConnected`) and the countability of the isolated points of a subspace (`countable_setOf_isolated_subtype`). This packages the mathlib PR leanprover-community/mathlib4#41918 as a project auxiliary file; it is a shared dependency of the variation-process and Doob-maximal-inequality developments. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
CoolRmal
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July 22, 2026 15:44
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BrownianMotion/Auxiliary/SeparableSpace.lean, proving that a separable linearly ordered topological space is hereditarily separable.Main results
exists_isOpen_ordConnected_mem_subset,isTopologicalBasis_isOpen_ordConnected,nhds_basis_isOpen_ordConnected— the open order-connected sets form a topological basis in a linearly ordered topological spacecountable_setOf_isolated_subtype— in a separable LOTS, the points of a subset that are isolated in the subspace topology form a countable set.Set.separableSpace : SeparableSpace ↥s— every subset of a separable LOTS is separable in the subspace topology.This PR is used in both #498 and #494.
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