feat(Topology/Connected): local (path-)connectedness of products and pi types

[korbonits]

Add Prod and Pi instances for LocallyConnectedSpace and LocallyPathConnectedSpace. The pi instances come in two flavors: finite index type, or arbitrary index type with all factors (path-)connected — the extra assumption is necessary, since e.g. an infinite product of discrete spaces is not locally connected.

Follow-up to #40092.

[github-actions]

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[github-actions]

PR summary f16796ee11

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ Pi.locallyConnectedSpace
+ Pi.locallyConnectedSpace_of_finite
+ Pi.locallyConnectedSpace_of_finite_nonpreconnected
+ Pi.locallyPathConnectedSpace
+ Pi.locallyPathConnectedSpace_of_finite
+ Pi.locallyPathConnectedSpace_of_finite_nonpathconnected
+ Prod.locallyConnectedSpace
+ Prod.locallyPathConnectedSpace

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 f16796e).

• +8 new declarations
• −0 removed declarations

+Pi.locallyConnectedSpace
+Pi.locallyConnectedSpace_of_finite
+Pi.locallyConnectedSpace_of_finite_nonpreconnected
+Pi.locallyPathConnectedSpace
+Pi.locallyPathConnectedSpace_of_finite
+Pi.locallyPathConnectedSpace_of_finite_nonpathconnected
+Prod.locallyConnectedSpace
+Prod.locallyPathConnectedSpace

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit f16796ee11
Reference commit 11d11a11a6

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).

[korbonits]

LLM-generated

[CoolRmal]

I think here's a better design. May you first prove the following?

theorem Pi.locallyConnectedSpace_of_finite_nonpreconnected
    [∀ i, TopologicalSpace (X i)] [∀ i, LocallyConnectedSpace (X i)]
    (hfinite : {i | ¬PreconnectedSpace (X i)}.Finite) : LocallyConnectedSpace (∀ i, X i) := sorry

which is saying that if every X i is locally connected and the set of i such that X i is not preconnected is finite, then LocallyConnectedSpace (∀ i, X i). Pi.locallyConnectedSpace_of_finite and Pi.locallyConnectedSpace should then be easy corollaries.

You can then do something similar for locally path connected space.

[CoolRmal]

awaiting-author

[CoolRmal]

And maybe if you want you can also prove the following characterization of locally connectedness of the product: ∀ i, X i is locally connected if either it is empty or every X i is locally connected and {i | ¬PreconnectedSpace (X i)}.Finite. To prove this I think you need to show first that if f : X → Y is open continuous surjective and X is locally connected, then so is Y.

[korbonits]

And maybe if you want you can also prove the following characterization of locally connectedness of the product: ∀ i, X i is locally connected if either it is empty or every X i is locally connected and {i | ¬PreconnectedSpace (X i)}.Finite. To prove this I think you need to show first that if f : X → Y is open continuous surjective and X is locally connected, then so is Y.

Is it OK if I do this in a follow-up PR or would you prefer to see it in this PR? I can commit to that as a fast follow. I think the helper lemma deserves its own visibility! Open to your thoughts here :) @CoolRmal

[CoolRmal]

I think it is fine to include these lemmas in this single PR, but maybe you can change the PR description a little bit.

[grunweg]

Hello from triage! Can you comment on whether you used AI for this project (and if so, label with PR with LLM-generated)? Thanks! (Per mathlib's AI policy, this is not forbidden in principle, but its usage must be disclosed.)