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Many mathematical structures come in symmetric and asymmetric versions. Classical examples include commutative and noncommutative algebraic structures, as well as symmetric preorders (=equivalence relations) and asymmetric such (usually partial orders). In these cases, there is always a duality available, whose use simplifies their study, and which reduces to the identity in the symmetric case. Also, in each of these cases, while symmetry is a simplifying assumption, there are many useful asymmetric examples. A similar phenomenon occurs in general topology, although in this case there are often many available useful duals. There are also many useful asymmetric spaces, such as the finite T0 spaces and the unit interval with the upper, or lower topology (in fact the Scott and lower topologies on any continuous lattice). The latter, using a dual, gives rise to the usual topology and order on the unit interval. |
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