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feat(Topology): An open cover of a preconnected set has no isolated components#39691

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feat(Topology): An open cover of a preconnected set has no isolated components#39691
Raph-DG wants to merge 1 commit into
leanprover-community:masterfrom
Raph-DG:Raph-DG-TransGen

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@Raph-DG Raph-DG commented May 22, 2026

In this PR we show that a family of open sets whose union is preconnected has the property that for any two (nonempty) sets U and V in the cover there is a sequence of sets in the cover U = U_0, U_1,..., U_n = V such that U_i \cap U_{i+1} is nonempty for all i.


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PR summary 04cea0add8

Import changes for modified files

No significant changes to the import graph

Import changes for all files
Files Import difference

Declarations diff

+ IsPreconnected.transGen_of_iUnion

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.


No changes to strong technical debt.
No changes to weak technical debt.

@github-actions github-actions Bot added the t-topology Topological spaces, uniform spaces, metric spaces, filters label May 22, 2026
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