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: 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) .
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 .

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