This is a follow up post to my previous post on . Mathematicians will often write without being clear of the context and structure associated with it.
To a Euclidean geometer means a circle – a maximal set of points equidistant from a given point. All circles are equivalent in the sense that they can be made equal by a translation and a scaling about the centre.
In a two dimensional inner product space typically means the set of all points with norm 1. A circle is more generally the set of all points with norm r for some real number r and is related to by a scaling transformation.
To a group theorist would mean the one dimensional orthogonal group – the group of all transformations in the plane.
To a complex analyst could mean either the set of points in with length 1 or it could be the group of linear transformations associated with multiplication by elements of this set. (These are quite different – in the set 1 has no special meaning but in the group it corresponds to the identity).
To a topologist means anything homeomorphic to a Euclidean circle – so includes ellipses, polygons, simple closed curves,…
To a differential geometer means anything diffeomorphic to a Euclidean circle, which doesn’t include “most” things a topologist means.
A set theorist wouldn’t call it , but to her it would be any set with the same cardinality as the real numbers.
This is a major theme of category theory – it’s not only the objects that matter but also the maps that preserve them – whether it be affine transformations, orthogonal transformations, group homomorphisms, group homomorphisms, homeomorphisms, diffeomorphisms or bijections. So if you must write to represent a structure at least make sure it is clear which category you are working in – i.e. which maps preserve the structure.