feat(Topology/Compactness/CompactSystem): introduce compact Systems

[pfaffelh]

A compact system is a set systems with the property that, whenever a countable intersections of sets in the set system is empty, there is a finite subset of sets with empty intersection. These are needed e.g. in measure theory if one wants to show sigma-additivity of a set function on a ring.

• Main result: The set of sets which are either compact and closed, or univ, is a compact system.
• Introduce Dissipate s x := ⋂ y ≤ x, s y (Data/Set/Dissipate), which is parallel to Data/Set/Accumulate.

Co-authored-by: Rémy Degenne [email protected]

[github-actions]

PR summary 3bb3ee2af4

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Data.Set.Dissipate (new file)	609
Mathlib.Topology.Compactness.CompactSystem (new file)	713

---

Declarations diff

+ Dissipate
+ IsCompactSystem
+ IsPiSystem.iff_of_empty_mem
+ biInter_dissipate
+ directed_dissipate
+ dissipate_antitone
+ dissipate_bot
+ dissipate_def
+ dissipate_eq
+ dissipate_eq_empty
+ dissipate_of_piSystem
+ dissipate_subset
+ dissipate_subset_dissipate
+ dissipate_subset_iInter
+ dissipate_succ
+ dissipate_zero
+ exists_dissipate_eq_empty_iff
+ exists_dissipate_eq_empty_iff_of_directed
+ exists_subset_dissipate_of_directed
+ finite_of_empty
+ iInter_dissipate
+ iff_directed
+ iff_directed'
+ iff_isCompactSystem_of_or_empty
+ iff_isCompactSystem_of_or_univ
+ iff_nonempty_iInter
+ iff_nonempty_iInter_of_lt
+ mem_dissipate
+ mono
+ of_IsEmpty
+ of_isCompact
+ of_isCompact_isClosed
+ of_isCompact_isClosed_or_univ

You can run this locally as follows

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

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

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

---

No changes to technical debt.

You can run this locally as

./scripts/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).

[leanprover-community-bot-assistant]

This pull request has conflicts, please merge master and resolve them.

[RemyDegenne]

I'm closing this PR since it has been replaced by #25899