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Topological orderings are also closely related to the concept of a linear extension of a partial order in mathematics. A partially ordered set is just a set of objects together with a definition of the "≤" inequality relation, satisfying the axioms of reflexivity ( x ≤ x ), antisymmetry (if x ≤ y and y ≤ x then x = y ) and transitivity ...
In fact topological insulators are different from topologically ordered states defined in this article. Topological insulators only have short-ranged entanglements and have no topological order, while the topological order defined in this article is a pattern of long-range entanglement. Topological order is robust against any perturbations.
A topological space X is called orderable or linearly orderable [1] if there exists a total order on its elements such that the order topology induced by that order and the given topology on X coincide. The order topology makes X into a completely normal Hausdorff space. The standard topologies on R, Q, Z, and N are the order topologies.
Therefore, every graph with a topological ordering is acyclic. Conversely, every directed acyclic graph has at least one topological ordering. The existence of a topological ordering can therefore be used as an equivalent definition of a directed acyclic graphs: they are exactly the graphs that have topological orderings. [2]
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 finest such topology is the Alexandrov topology , given by ...
Total orders, orderings that specify, for every two distinct elements, which one is less than the other; Weak orders, generalizations of total orders allowing ties (represented either as equivalences or, in strict weak orders, as transitive incomparabilities) Well-orders, total orders in which every non-empty subset has a least element
In the field of computer science, a pre-topological order or pre-topological ordering of a directed graph is a linear ordering of its vertices such that if there is a directed path from vertex u to vertex v and v comes before u in the ordering, then there is also a directed path from vertex v to vertex u.
Three well-orderings on the set of natural numbers with distinct order types (top to bottom): , +, and +. Every well-ordered set is order-equivalent to exactly one ordinal number , by definition. The ordinal numbers are taken to be the canonical representatives of their classes, and so the order type of a well-ordered set is usually identified ...