A Note on Compactness and Clique Size

Abstract

Say that a topological space $X$ has finite, respectively bounded, cliques iff every closed, irreflexive binary relation --- equivalently, every closed, loop-free directed graph --- on $X$ has cliques of finite, respectively bounded finite, size. Every compact space has bounded cliques. Having finite cliques implies limit-point compactness, and is implied by $ω$-limit point compactness (equivalently, countable compactness). Thus, for $T_1$ spaces, having finite cliques is equivalent to countable compactness. Having bounded cliques is strictly weaker than compactness. Indeed, any space $X$ such that $X^ω$ is countably compact has bounded cliques. However, we have found no example of a countably compact space having finite but unbounded cliques. The existence of such a space is the major open problem raised in this note.

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“14 pages. Assisted by Anthropic's Claude; formalized in Lean 4 and machine-checked”

arXiv metadata: comment

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Code generation, completion, or debugging

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2

Verified

2026-08-23T05:13:36.370673Z

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