Means on chainable continua

Source abstract

By a mean on a space X X we understand a mapping μ : X × X → X \mu :X\times X\to X such that μ ( x , y ) = μ ( y , x ) \mu (x,y)=\mu (y,x) and μ ( x , x ) = x \mu (x,x)=x for x , y ∈ X x,y\in X . A chainable continuum is a metric compact connected space which admits an ε \varepsilon - mapping onto the interval [ 0 , 1 ] [0,1] for every number ε > 0 \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.