bibkey: gleason1958projective
authors: Andrew M. Gleason
year: 1958
title: Projective topological spaces
doi: 10.1215/ijm/1255454110
url: https://projecteuclid.org/euclid.ijm/1255454110
claim: Compact Hausdorff projective spaces are exactly the extremally disconnected spaces.
strata_touched: []
license: citation-only
triage: anchor

Projective compact Hausdorff covers and dynamics

Gleason, Illinois Journal of Mathematics 2, pp.482–489, Theorem 2.5, characterizes projective compact Hausdorff spaces by extremal disconnectedness. A space is extremally disconnected when the closure of each open set is open. Projectivity guarantees continuous lifting through compact Hausdorff surjections. It does not assert shadowing of the lift.

The pinned Mathlib module Mathlib.Topology.ExtremallyDisconnected attributes and proves this characterization as CompactT2.projective_iff_extremallyDisconnected; its direction CompactT2.Projective.extremallyDisconnected is enough for the exclusion below. Its definition and closure lemmas, together with the repository’s finite_observation_isSofic, supply the prior machinery for D5/S1/Dynamics/Shadowing/StoneanForwardTubeObstruction.

For the successor map on the compact convergent sequence of natural numbers, every compact Hausdorff projective cover with any continuous equivariant lift fails shadowing. The isolated-point fibers are nonempty disjoint open forward tubes; extreme disconnectedness supplies their arbitrary clopen coloring, while shadowing would force a finite graph observation language. The power-of-two coloring contradicts that graph language by whole-path cycle erasure. The exact proof and its boundary are in the owning problem dossier and Blueprint.

This is a construction obstruction, not a negative answer to Peng’s Question 5.1: the convergent-sequence source itself is a zero-dimensional shadowing cover. No attribution of this dynamic obstruction to Gleason, and no worldwide novelty claim for it, is made.

Verified locator

Gleason, Projective topological spaces, Illinois J. Math. 2, pp.482–489, Theorems 1.2 and 2.5.

• DOI: https://doi.org/10.1215/ijm/1255454110
• Publisher: https://projecteuclid.org/euclid.ijm/1255454110