feat(Topology/GDelta): a completely metrizable subspace of a metrizable space is Gδ

[justin-palumbo]

Adds TopologicalSpace.IsCompletelyMetrizableSpace.isGδ: a completely metrizable subspace of a metrizable space is Gδ

This is one direction of a foundational fact in descriptive set theory --- the Polish subspaces of a Polish space are exact the Gδ subset.

The proof here is modeled on the exposition in Anush Tserunyan's descriptive set theory notes https://www.math.mcgill.ca/atserunyan/Teaching_notes/dst_lectures.pdf, with one adjustment; to make the formalization easier and avoid having multiple distance metrics instantiated at the same time, instead of directly defining open U_n with s = ⋂ n, U define them such that s = closure s ∩ ⋂ n, U n and then appeal to the fact closed sets are Gδ in a perfectly normal space.

---

• depends on: fix(Topology/Metrizable/CompletelyMetrizable): make instances anonymous to avoid name collisions #42896 [which fixes buggy naming present in Topology/Metrizable/CompletelyMetrizable]

This can be viewed as a companion to #42693, which - under the stronger assumption of the ambient space being Polish - establishes the other direction --- a Gδ subspace of a Polish space is itself Polish.

Together, these give Alexandrov's theorem, characterizing the Polish subspaces of an ambient Polish space as exactly the Gδ ones, and (via Metric.PiNatEmbed.exists_embedding_to_hilbert_cube) Polish spaces as exactly the Gδ subsets of the Hilbert cube. If and when both these PRs are committed, I anticipate adding these consequences to Topology/MetricSpace/Polish.lean

LLM usage: I wrote the initial version myself, while using Google's free "AI mode" to help me find relevant lemma names in the codebase, and afterward used helped Claude Opus to review, which led to simplifying some of the arguments. I take full responsibility for the code here

[github-actions]

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[github-actions]

PR summary 0825c476b5

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Topology.GDelta.CompletelyMetrizable (new file)	1157

---

Declarations diff (regex)

+ instance (priority := 90) [TopologicalSpace X]
+ instance (priority := 90) [TopologicalSpace X] [IsCompletelyMetrizableSpace X] :
+ isGδ
- instance (priority := 90) MetrizableSpace [TopologicalSpace X] [IsCompletelyMetrizableSpace X] :
- instance (priority := 90) PseudoMetrizableSpace [TopologicalSpace X]

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 0825c47).

• +3 new declarations
• −2 removed declarations

-TopologicalSpace.IsCompletelyMetrizableSpace.MetrizableSpace
+TopologicalSpace.IsCompletelyMetrizableSpace.instMetrizableSpace
+TopologicalSpace.IsCompletelyMetrizableSpace.isGδ
-TopologicalSpace.IsCompletelyPseudoMetrizableSpace.PseudoMetrizableSpace
+TopologicalSpace.IsCompletelyPseudoMetrizableSpace.instPseudoMetrizableSpace

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 0825c476b5
Reference commit 618f225e1f

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).

[github-actions]

✅ PR Title Formatted Correctly

The title of this PR has been updated to match our commit style conventions.
Thank you!

[mathlib-dependent-issues]

This PR/issue depends on:

• fix(Topology/Metrizable/CompletelyMetrizable): make instances anonymous to avoid name collisions #42896
  By Dependent Issues (🤖). Happy coding!