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feat(Topology): A spectral map between quasi-separated, prespectral sober spaces has compact fibers#39663

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feat(Topology): A spectral map between quasi-separated, prespectral sober spaces has compact fibers#39663
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Raph-DG:Raph-DG-Constructible

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@Raph-DG Raph-DG commented May 21, 2026

In this PR, we develop some API around the constructible topology, culminating in the fact that a spectral map between quasi-separated, prespectral sober spaces has compact fibers. To see why this might be of interest, note that the analogous theorem in algebraic geometry (that a quasiseparated map between schemes has quasicompact fibers) does not require any global separatedness assumptions, and the proof of this is very algebraic. So we have the somewhat mysterious situation that it seems as though there are nontrvial topological restrictions on the kinds of spectral maps which can be the underlying maps of morphisms of schemes.

This PR was originally part of #26304, a PR on pushforwards of algebraic cycles. This is where the notion of compactness of fibers becomes relevant, as this guarantees each coefficient of the pushforward of a cycle is computed by a finite sum.


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Raph-DG and others added 26 commits April 10, 2026 11:26
This reverts commit bb5278f.
Co-authored-by: Christian Merten <christian@merten.dev>
Co-authored-by: Christian Merten <christian@merten.dev>
Co-authored-by: Christian Merten <christian@merten.dev>
Co-authored-by: Christian Merten <christian@merten.dev>
Co-authored-by: Christian Merten <christian@merten.dev>
Co-authored-by: Christian Merten <christian@merten.dev>
Co-authored-by: Christian Merten <christian@merten.dev>
@github-actions github-actions Bot added the t-topology Topological spaces, uniform spaces, metric spaces, filters label May 21, 2026
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github-actions Bot commented May 21, 2026

PR summary b6865bbcda

Import changes for modified files

Dependency changes

File Base Count Head Count Change
Mathlib.Topology.Spectral.ConstructibleTopology 890 893 +3 (+0.34%)
Import changes for all files
Files Import difference
Mathlib.Topology.Spectral.ConstructibleTopology 3

Declarations diff

+ HasCompactFibers
+ IsOpenEmbedding.isSpectralMap_of_compactSpace
+ IsProperMap.hasCompactFibers
+ IsSpectralMap.hasCompactFibers
+ WithConstructibleTopology.continuous_equiv
+ WithConstructibleTopology.equiv
+ WithConstructibleTopology.map
+ WithConstructibleTopology.map_comp
+ WithConstructibleTopology.map_continuous
+ WithConstructibleTopology.map_id
+ instance [PrespectralSpace X] [T0Space X] [QuasiSober X] :
+ isOpen_ofConstructibleTopology_preimage_iff
+ toConstructibleTopology

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.

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