feat(Topology/Order): set of compact elements, generating basic opens contained in a Locale.PT, is directed

[edwin1729]

This is (2/4) PRs culminating in a proof that "Scott Topologies over Algebraic DCPOs are Sober". In this PR we prove properties of a certain set of compact elements, w.r.t a Locale.PT.

A Locale.PT can be thought of as a set of Opens. Precisely it is a Frame homomorphism from Opens _ to Prop so we are essentially choosing certain Opens to form our set and this set has additional properties. Namely it is a completely prime filter: if the supremum of some Opens is contained than atleast one of the Opens must be contained.

Take some Locale.PT, x. In this PR, we prove that the set of compact elements generating basic opens (compact elements generate topological basis see #31662) contained in x is directed.

This set of compact elements defined above is important in subsequent parts of the proof since it fully determines x.

• depends on: feat(Topology/Order): topological basis of scott topology on Complete… #31662

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[github-actions]

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[github-actions]

PR summary d4d96de5f1

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Topology.Order.ScottTopologyDCPO (new file)	1643

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Declarations diff

+ AlgebraicDCPO
+ CompactElement
+ CompactElement.toOpen
+ IsBasis.open_eq_sSup
+ IsCompactElement.toOpen
+ directed_setOf_compactElement
+ exists_basis_mem_basis
+ isCompactElement_iff_le_of_directed_sSup_le
+ isOpen_of_basis
+ isTopologicalBasis_Ici_image_compactSet
+ nonempty_setOf_compactElement
+ specialization_iff_ge

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.

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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-dependent-issues]

This PR/issue depends on:

• feat(Topology/Order): topological basis of scott topology on Complete… #31662
  By Dependent Issues (🤖). Happy coding!