feat(Topology/Order): separability is hereditary for linearly ordered topological spaces

[CoolRmal]

In this PR we prove that

1. isTopologicalBasis_isOpen_ordConnected: in a linearly ordered topological space, every point of an open set U has an open Set.OrdConnected neighbourhood contained in U.
2. countable_setOf_isolated_subtype: in a separable linearly ordered topological space, the points of a subset s that are isolated in the subspace s form a countable set.
3. A subset of a separable linearly ordered topological space is separable in the subspace topology. This is proved as an instance.

The material was developed for showing the measurability of the pathwise variation of a continuous stochastic process in RemyDegenne/brownian-motion#494. Created with the help of Claude Code, carefully reviewed and golfed by myself.

[github-actions]

PR summary 73788e18da

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Topology.Order.Separable (new file)	875

---

Declarations diff (regex)

+ Dense.mem_of_isOpen_singleton
+ Set.separableSpace
+ countable_isolated_inter_Ioi_aux
+ countable_setOf_isolated_subtype
+ exists_isOpen_ordConnected_mem_subset
+ isTopologicalBasis_isOpen_ordConnected
+ nhds_basis_isOpen_ordConnected

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 73788e1).

• +6 new declarations
• −0 removed declarations

+Dense.mem_of_isOpen_singleton
+Set.separableSpace
+countable_setOf_isolated_subtype
+exists_isOpen_ordConnected_mem_subset
+isTopologicalBasis_isOpen_ordConnected
+nhds_basis_isOpen_ordConnected

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 73788e18da
Reference commit e780b56e92

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[felixpernegger]

(I haven't looked at the lean code)
The main results can be generalised; we only need to assume that X is separaböe and the subset of a linear odered topology (not necessarily seperable) for it to be hereditarily separable. See https://topology.pi-base.org/theorems/T000294

(many space are subspaces of order topologies but not order topologies themselves, i.e. (0,1) \cup {2})

I think its fine if you dont want to generalise it now; but maybe note it as TODO somewhere

[CoolRmal]

I am aware of the GO-spaces but I am not very motivated to define a typeclass for them now. I just included a TODO and also opened an issue: #42275 (I guess this would also be a good issue for beginners).

Once a typeclass for the GO-space is defined it should be easy to generalize the result, as my proof really only use the property that IsTopologicalBasis {V : Set α | IsOpen V ∧ V.OrdConnected}, which is basically how GO-space is defined.