Mean operation
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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, http://eudml.org/doc/160126.
- 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), doi:10.2478/taa-2014-0002, http://eudml.org/doc/266591.
- Charatonik, Janusz J. (2003), "Selected problems in continuum theory", Topology Proceedings 27 (1): 51–78, http://topology.nipissingu.ca/tp/reprints/v27/tp27107.pdf.
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