feat(Topology/MetricSpace/Heine-Cantor): metric ε-δ forms of Heine–Cantor

[teorth]

Add (pseudo-)metric space (or "epsilon-delta") versions of many of the versions of the Heine-Cantor theorem (continuous functions on compacta are uniformly continuous) from Topology.UniformSpace.HeineCantor into a new file
Topology.Metric.HeineCantor.

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A new file needed to be created because many files downstream of this file blocked the use of UniformOnFun.

AI assistance was used to expand a small initial set of human-generated lemmas into a larger set, with the latter then reviewed by both an AI tool and myself.

[github-actions]

PR summary 598bbec437

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Topology.MetricSpace.HeineCantor (new file)	1025

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Declarations diff (regex)

+ Continuous.exists_forall_dist_le_of_compactSpace
+ Continuous.exists_forall_dist_lt_of_compactSpace
+ ContinuousOn.exists_forall_dist_le_of_isCompact
+ ContinuousOn.exists_forall_dist_lt_of_isCompact
+ IsCompact.exists_forall_dist_le_of_continuousAt
+ IsCompact.exists_forall_dist_lt_of_continuousAt

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 598bbec).

• +6 new declarations
• −0 removed declarations

+Continuous.exists_forall_dist_le_of_compactSpace
+Continuous.exists_forall_dist_lt_of_compactSpace
+ContinuousOn.exists_forall_dist_le_of_isCompact
+ContinuousOn.exists_forall_dist_lt_of_isCompact
+IsCompact.exists_forall_dist_le_of_continuousAt
+IsCompact.exists_forall_dist_lt_of_continuousAt

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 598bbec437
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).

[github-actions]

✅ PR Title Formatted Correctly

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