feat(Topology/Sets): define the Vietoris topology on (Nonempty)Compacts

[gasparattila]

This PR defines the Vietoris topology on Compacts and NonemptyCompacts, and proves that it is induced by the Hausdorff uniformity.

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• depends on: [Merged by Bors] - feat(Topology/UniformSpace): define the Hausdorff uniformity #31031
• depends on: [Merged by Bors] - feat: SetRel.image and SetRel.preimage are open #31058
• depends on: feat(Topology/Sets): range of NonemptyCompacts.toCompacts #34036

[github-actions]

PR summary e2a36edc3e

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Topology.UniformSpace.Closeds	683	685	+2 (+0.29%)

Import changes for all files

Files	Import difference
Mathlib.Topology.MetricSpace.Closeds Mathlib.Topology.MetricSpace.GromovHausdorff	1
Mathlib.Topology.UniformSpace.Closeds	2
Mathlib.Topology.Sets.VietorisTopology (new file)	667

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Declarations diff

+ IsCompact.nhds_hausdorff_eq_nhds_vietoris
+ TotallyBounded.exists_prodMk_finset_mem_hausdorffEntourage
+ TotallyBounded.nhds_vietoris_le_nhds_hausdorff
+ _root_.IsClosed.powerset_vietoris
+ _root_.IsOpen.powerset_vietoris
+ coe_eq_empty
+ coe_nonempty
+ isClopen_singleton_bot
+ isClopen_singleton_empty
+ isClosedEmbedding_toCompacts
+ isOpenEmbedding_toCompacts
+ preimage_eq_image
+ range_toCompacts
+ vietoris
++ continuous_coe
++ isEmbedding_coe
++ isOpen_subsets_of_isOpen
++ topology
+++ isClosed_inter_nonempty_of_isClosed
+++-- isOpen_inter_nonempty_of_isOpen
++-- isClosed_subsets_of_isClosed

You can run this locally as follows

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

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

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

---

No changes to technical debt.

You can run this locally as

./scripts/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).

[mathlib4-merge-conflict-bot]

This pull request has conflicts, please merge master and resolve them.

[mathlib4-merge-conflict-bot]

This pull request has conflicts, please merge master and resolve them.

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• [Merged by Bors] - feat(Topology/UniformSpace): define the Hausdorff uniformity #31031
• [Merged by Bors] - feat: SetRel.image and SetRel.preimage are open #31058
• feat(Topology/Sets): range of NonemptyCompacts.toCompacts #34036
  By Dependent Issues (🤖). Happy coding!

[mathlib4-merge-conflict-bot]

This pull request has conflicts, please merge master and resolve them.