Means on chainable continua

Abstract:

By a mean on a space $X$ we understand a mapping $\mu :X\times X\to X$ such that $\mu (x,y)=\mu (y,x)$ and $\mu (x,x)=x$ for $x,y\in X$. A chainable continuum is a metric compact connected space which admits an $\varepsilon$- mapping onto the interval $[0,1]$ for every number $\varepsilon >0$. We show that every chainable continuum that admits a mean is homeomorphic to the interval. In this way we answer a question by P. Bacon. We answer some other questions concerning means as well.