feat: typeclass for zero-dimensional spaces

[vihdzp]

---

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

[github-actions]

PR summary 4ec5d95aa6

Import changes exceeding 2%

%	File
+14.08%	Mathlib.Topology.UniformSpace.Ultra.Basic

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Topology.UniformSpace.Ultra.Basic	618	705	+87 (+14.08%)
Mathlib.Topology.Separation.CompletelyRegular	1550	1552	+2 (+0.13%)
Mathlib.Topology.Separation.Lemmas	1554	1555	+1 (+0.06%)

Import changes for all files

Files	Import difference
5 files Mathlib.MeasureTheory.Measure.DiracProba Mathlib.MeasureTheory.Measure.LevyConvergence Mathlib.MeasureTheory.Measure.Prokhorov Mathlib.Probability.CentralLimitTheorem Mathlib.Topology.Separation.Lemmas	1
5 files Mathlib.Topology.MetricSpace.Snowflaking Mathlib.Topology.Separation.CompletelyRegular Mathlib.Topology.UniformSpace.Ultra.Completion Mathlib.Topology.UniformSpace.Ultra.Constructions Mathlib.Topology.UniformSpace.Uniformizable	2
Mathlib.Topology.UniformSpace.Ultra.Basic	87

---

Declarations diff

+ CompletelyRegularSpace.of_isTopologicalBasis_clopens
+ CompletelyRegularSpace.totallySeparatedSpace_of_cardinalMk_lt_continuum
+ CompletelyRegularSpace.zeroDimensionalSpace_of_cardinalMk_lt_continuum
+ IndiscreteTopology.isClopen_iff
+ ZeroDimensionalSpace
+ ZeroDimensionalSpace.of_isOpen_of_nhds
+ exists_isClopen_mem_of_isOpen
+ hasBasis_nhds_isClopen
+ instance : ZeroDimensionalSpace X := by
+ instance [Countable X] [T35Space X] : TotallySeparatedSpace X
+ instance [DiscreteTopology X] : ZeroDimensionalSpace X := by
+ instance [IndiscreteTopology X] : ZeroDimensionalSpace X := by
+ instance [T0Space X] [ZeroDimensionalSpace X] : TotallySeparatedSpace X := by
+ instance [T2Space X] [CompactSpace X] [TotallyDisconnectedSpace X] : ZeroDimensionalSpace X := by
+ instance [TotallyDisconnectedSpace H] : ZeroDimensionalSpace H := by
+ instance [ZeroDimensionalSpace X] : CompletelyRegularSpace X
+ totallyDisconnectedSpace_iff_totallySeparatedSpace
- _root_.Set.Countable.totallySeparatedSpace
- _root_.TopologicalSpace.isTopologicalBasis_clopens
- instance (priority := 100) [TotallyDisconnectedSpace H] : TotallySeparatedSpace H
- instance [Countable X] : TotallySeparatedSpace X
- totallySeparatedSpace_of_cardinalMk_lt_continuum

You can run this locally as follows

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

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

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

---

Increase in tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type
454	1	porting notes

Current commit 11eee01e7e
Reference commit 4ec5d95aa6

You can run this locally as

./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).

[mathlib-dependent-issues]

This PR/issue depends on:

• feat: a topology with a basis of clopens is completely regular #37317
  By Dependent Issues (🤖). Happy coding!

[FernandoChu]

I wonder, what happens with the class ZeroDimensionalSpace when we get the a typeclass for IsDimensionalSpace n?

[vihdzp]

I figured it could be similar to how we have Ring.dimensionLEOne but also Ring.krullDimLE?

[FernandoChu]

I see. If you're ok with it I can implement HasSmallInductiveDimensionLE so that we don't have this situation.

[vihdzp]

Go for it! As long as the definition doesn't use too many imports, and as long as we have HasSmallInductiveDimensionLE 0 iff basis of clopens.