Compactifications of the ray with the arc as remainder admit no 𝑛-mean

Abstract

An n-mean on X is a function F : X n → X F:{X^n} \to X which is idempotent and symmetric. In 1970 P. Bacon proved that the sin ⁡ ( 1 / x ) \sin (1/x) continuum admits no 2-mean. In this paper, it is proved that if X is any metric space which contains an open line one of whose boundary components in X is an arc, then X admits no n-mean, n ≥ 2 n \geq 2 .