refactor(Topology): rename LocPathConnected

[felixpernegger]

Rename LocPathConnected to LocallyPathConnected per discussion at https://leanprover.zulipchat.com/#narrow/channel/287929-mathlib4/topic/Rename.20.60.60.60LocPathConnectedSpace.60.60.60.3F/with/582824006

Deprecations were partly added with Claude

---

[felixpernegger]

-t-differential-geometry

[github-actions]

PR summary ea0501174b

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Topology.Connected.LocPathConnected	1
Mathlib.Topology.Connected.LocallyPathConnected (new file)	1199

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Declarations diff

+ AlexandrovDiscrete.locallyPathConnectedSpace
+ ChartedSpace.locallyPathConnectedSpace
+ Convex.locallyPathConnectedSpace
+ IsOpen.locallyPathConnectedSpace
+ LocConvexSpace.toLocallyPathConnectedSpace
+ LocallyPathConnectedSpace
+ LocallyPathConnectedSpace.coinduced
+ LocallyPathConnectedSpace.of_bases
+ PathConnectedSpace.of_locallyPathConnectedSpace
+ Quot.locallyPathConnectedSpace
+ Quotient.locallyPathConnectedSpace
+ Sigma.locallyPathConnectedSpace
+ Sum.locPathConnectedSpace
+ Sum.locallyPathConnectedSpace.{u}
+ Topology.IsOpenEmbedding.locallyPathConnectedSpace
+ Topology.IsQuotientMap.locallyPathConnectedSpace
+ Unitary.instLocallyPathConnectedSpace
+ instLocPathConnectedSpace
+ instance (priority := 100) LocallyConvexSpace.toLocallyPathConnectedSpace [Module ℝ E]
+ instance (priority := 100) instLocallyPathConnectedSpace : LocallyPathConnectedSpace X
+ instance : LocallyPathConnectedSpace (EuclideanQuadrant n)
+ instance : LocallyPathConnectedSpace ℍ := isOpenEmbedding_coe.locallyPathConnectedSpace
+ instance [DeltaGeneratedSpace X] : LocallyPathConnectedSpace X := by
+ instance [Group G] [IsTopologicalGroup G] [LocallyPathConnectedSpace G] :
+ instance [NeZero n] : LocallyPathConnectedSpace (EuclideanHalfSpace n)
+ locallyPathConnectedSpace_iff_isOpen_pathComponentIn
+ locallyPathConnectedSpace_iff_pathComponentIn_mem_nhds
- Sum.locPathConnectedSpace.{u}
- instance (priority := 100) LocallyConvexSpace.toLocPathConnectedSpace [Module ℝ E]
- instance (priority := 100) instLocPathConnectedSpace : LocPathConnectedSpace X
- instance : LocPathConnectedSpace (EuclideanQuadrant n)
- instance : LocPathConnectedSpace ℍ := isOpenEmbedding_coe.locPathConnectedSpace
- instance [DeltaGeneratedSpace X] : LocPathConnectedSpace X := by
- instance [Group G] [IsTopologicalGroup G] [LocPathConnectedSpace G] :
- instance [NeZero n] : LocPathConnectedSpace (EuclideanHalfSpace n)
-+-+ existsUnique_continuousMap_lifts

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./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).

[mathlib-bors]

This pull request is now in draft mode. No active bors state needed cleanup.

While this PR remains draft, bors will ignore commands on this PR. Mark it ready for review before using commands like bors r+ or bors try.

[felixpernegger]

t-topology