Topology: pseudometrizable spaces are paracompact

[CoolRmal]

This PR proves Stone’s theorem that every pseudometrizable space is paracompact. It installs a compatible pseudometric using the existing PseudoMetrizableSpace API and transfers the established paracompactness theorem for pseudoemetric spaces.

It also records the resulting T4 theorem for metrizable spaces. This resolves the generalization TODO in Topology/EMetricSpace/Paracompact without changing its existing public API.

This is an independent metrizable-space PR associated with the Kupka--Prikry formalization project.

Checks:

• lake build
• lake exe lint-style Mathlib.Topology.Metrizable.Paracompact
• git diff --check upstream/master...HEAD

[github-actions]

PR summary 8c310991fb

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Topology.Metrizable.Paracompact (new file)	1145

---

Declarations diff (regex)

+ MetrizableSpace.t4Space
+ instance (priority := 100) PseudoMetrizableSpace.paracompactSpace

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 8c31099).

• +2 new declarations
• −0 removed declarations

+TopologicalSpace.MetrizableSpace.t4Space
+TopologicalSpace.PseudoMetrizableSpace.paracompactSpace

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 8c310991fb
Reference commit 69b866d83a

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).

[CoolRmal]

Closing this draft: the project does not want a separate pseudometrizable-paracompactness wrapper PR.