Means on scattered compacta

TL;DR

This paper proves that certain topological spaces containing a specific uncountable subset cannot support continuous mean operations or diagonally continuous n-means, highlighting limitations in their algebraic structure.

Contribution

It establishes the non-existence of separately continuous means on spaces with a cocountable subset homeomorphic to [0,ω₁], extending understanding of topological algebra constraints.

Findings

No separately continuous mean operation exists on such spaces.

No diagonally continuous n-mean exists for n ≥ 2 in these spaces.

Results highlight limitations in algebraic structures on certain topological spaces.

Abstract

We prove that a separable Hausdorff topological space $X$ containing a cocountable subset homeomorphic to $[0,\omega_1]$ admits no separately continuous mean operation and no diagonally continuous $n$-mean for $n\ge 2$.