Editing Order theory (section)

=== Topology ===
In [[topology]], orders play a very prominent role. In fact, the collection of [[open set]]s provides a classical example of a complete lattice, more precisely a complete [[Heyting algebra]] (or "'''frame'''" or "'''locale'''"). [[filter (mathematics)|Filters]] and [[net (mathematics)|nets]] are notions closely related to order theory and the [[closed set|closure operator of sets]] can be used to define a topology. Beyond these relations, topology can be looked at solely in terms of the open set lattices, which leads to the study of [[pointless topology]]. Furthermore, a natural preorder of elements of the underlying set of a topology is given by the so-called [[specialization order]], that is actually a partial order if the topology is [[T0 space|T<sub>0</sub>]].

Conversely, in order theory, one often makes use of topological results. There are various ways to define subsets of an order which can be considered as open sets of a topology. Considering topologies on a poset (''X'', ≤) that in turn induce ≤ as their specialization order, the [[Comparison of topologies|finest]] such topology is the [[Alexandrov topology]], given by taking all upper sets as opens. Conversely, the [[Comparison of topologies|coarsest]] topology that induces the specialization order is the [[upper topology]], having the complements of [[ideal (order theory)|principal ideals]] (i.e. sets of the form {''y'' in ''X'' | ''y'' ≤ ''x''} for some ''x'') as a [[subbase]]. Additionally, a topology with specialization order ≤ may be [[Specialization (pre)order#Important properties|order consistent]], meaning that their open sets are "inaccessible by directed suprema" (with respect to ≤). The finest order consistent topology is the [[Scott topology]], which is coarser than the Alexandrov topology. A third important topology in this spirit is the [[Lawson topology]]. There are close connections between these topologies and the concepts of order theory. For example, a function preserves directed suprema if and only if it is [[continuous function (topology)|continuous]] with respect to the Scott topology (for this reason this order theoretic property is also called [[Scott-continuous|Scott-continuity]]).