Mean operation

In algebraic topology, a mean or mean operation on a topological space X is a continuous, commutative, idempotent binary operation on X. If the operation is also associative, it defines a semilattice. A classic problem is to determine which spaces admit a mean. For example, Euclidean spaces admit a mean -- the usual average of two vectors -- but spheres of positive dimension do not, including the circle.

Further reading

• Aumann, G. (1943), "Über Räume mit Mittelbildungen.", Mathematische Annalen, 119 (2): 210–215, doi:10.1007/bf01563741.
• Sobolewski, Mirosław (2008), "Means on chainable continua", Proceedings of the American Mathematical Society, 136 (10): 3701–3707, doi:10.1090/s0002-9939-08-09414-8.
• T. Banakh, W. Kubis, R. Bonnet (2014), "Means on scattered compacta", Topological Algebra and its Applications, 2 (1), arXiv:1309.2401, doi:10.2478/taa-2014-0002{{citation}}: CS1 maint: multiple names: authors list (link).
• Charatonik, Janusz J. (2003), "Selected problems in continuum theory" (PDF), Proceedings of the Spring Topology and Dynamical Systems Conference, Topology Proceedings, 27 (1): 51–78, MR 2048922.