Asymmetry and duality in topology

Abstract

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 T 0 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.