feat(Topology/Metrizable): generalize compact separability

[yuanyi-350]

migrate from #42470

[github-actions]

PR summary d0e503664b

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Topology.EMetricSpace.Basic	1032	1034	+2 (+0.19%)

Import changes for all files

Files	Import difference
7 files Mathlib.Analysis.Oscillation Mathlib.Topology.EMetricSpace.Diam Mathlib.Topology.EMetricSpace.Lipschitz Mathlib.Topology.EMetricSpace.Pi Mathlib.Topology.MetricSpace.Pseudo.Pi Mathlib.Topology.MetricSpace.Pseudo.Real Mathlib.Topology.MetricSpace.Ultra.Pi	1
4 files Mathlib.Topology.EMetricSpace.Basic Mathlib.Topology.EMetricSpace.Paracompact Mathlib.Topology.MetricSpace.Cauchy Mathlib.Topology.UniformSpace.Real	2

---

Declarations diff (regex)

++- subset_countable_closure_of_compact

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 d0e5036).

• +1 new declarations
• −0 removed declarations

+TopologicalSpace.subset_countable_closure_of_compact

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit d0e503664b
Reference commit 863e60949d

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).