A topological proof that compact Hausdorff spaces are not finitely co-concrete

Marco Abbadini

arXiv:2607.29103v1 Primary: math.CT Secondary: math.GN

Abstract

Lieberman, Rosický, and Vasey proved that $\mathbf{CompHaus}^{\mathrm{op}}$ - the opposite of the category of compact Hausdorff spaces - is not finitely concrete by a route through Hilbert and Banach spaces, commutative unital $C^*$-algebras, and Gelfand duality. We give a short topological proof. Moreover, we strengthen the result by identifying a specific sequential colimit in $\mathbf{CompHaus}^{\mathrm{op}}$ that no faithful set-valued functor preserves.

Disclosure

“s a directed diagram. A faithful functor preserving all directed colimits would make κU bijective, whereas Theorem 3.1 shows that it is not surjective. □ Declaration of generative AI and AI-assisted technologies During the development of this work, I used OpenAI’s ChatGPT (ac- cessed in July 2026) to seek a direct topological proof that the category CompHausop is not finitely concrete. ChatGPT produced the fixed-diagram formulation”
PDF page 4

Classification: Substantial mathematical content or result generation
Multiplier: 10
Verified: 2026-08-23T19:15:46.543107Z

Structural counts

Pages 4 pdf

Theorems 1 source

Lemmas 1 source

Propositions 0 source

Corollaries 1 source

Definitions 1 source

Displayed equations 25 source

Bibliography entries 1 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file main.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.