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 $\omega$-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 $X^{\omega }$ 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.

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