In mathematics, specifically in order theory and functional analysis, the order topology of an ordered vector space is the finest locally convex topological vector spacetopology on X for which every order interval is bounded, where an order interval in X is a set of the form := where a and b belong to X. The order topology is an important topology that is used frequently in the theory of ordered topological vector spaces because the topology stems directly from the algebraic and order theoretic properties of, rather than from some topology that X starts out having. This allows for establishing intimate connections between this topology and the algebraic and order theoretic properties of. For many ordered topological vector spaces that occur in analysis, their topologies are identical to the order topology.
Definitions
The family of all locally convex topologies on X for which every order interval is bounded is non-empty and the order topology is the upper bound of this family. A subset of X is a neighborhood of 0 in the order topology if and only if it is convex and absorbs every order interval in X. Note that a neighborhood of 0 in the order topology is necessarily absorbing since = for all x in X. For every a ≥ 0, let and endow Xa with its order topology. The set of all Xa's is directed under inclusion and if Xa ⊆ Xb then the natural inclusion of Xa into Xb is continuous. If X is a regularly orderedvector space over the reals and if H is any subset of the positive coneC of X that is cofinal in C, then X with its order topology is the inductive limit of . The lattice structure can compensate in part for any lack of an order unit: In particular, if is an ordered Fréchet lattice over the real numbers then is the ordered topology on X if and only if the positive cone of X is a normal cone in. If X is a regularly ordered vector lattice then the ordered topology is the finest locally convex TVS topology on X making X into a locally convex vector lattice. If in addition X is order complete then X with the order topology is a barreled space and every band decomposition of X is a topological direct sum for this topology. In particular, if the order of a vector lattice X is regular then the order topology is generated by the family of all lattice seminorms on X.
Properties
Throughout we let be an ordered vector space and we let ?≤ denote the order topology on X.
If Xb separates points in X then is a bornological locally convex TVS.
Each positive linear operator between two ordered vector spaces is continuous for the respective order topologies.
Each order unit of an ordered TVS is interior to the positive cone for the order topology.
If the order of an ordered vector space X is a regular order and if each positive sequence of type inX is order summable, then X endowed with its order topology is a barreled space.
If the order of an ordered vector space X is a regular order and if for all x ≥ 0 and y ≥ 0 we have + = then the positive cone of X is a normal cone in X when X is endowed with the order topology.
If is an Archimedean ordered vector space over the real numbers having an order unit and let ?≤ denote the order topology on X. Then is an ordered TVS that is normable, ?≤ is the finest locally convex TVS topology on X such that the positive cone is normal, and the following are equivalent:
is complete.
Each positive sequence of type in X is order summable.
In particular, if is an Archimedean ordered vector space having an order unit then the order ≤ is a regular order and Xb = X+.
If X is a Banach space and an ordered vector space with an order unit then X's topological is identical to the order topology if and only if the positive cone of X is a normal cone in X.
If M is a solidvector subspace of a vector lattice X, then the order topology of X/M is the quotient of the order topology on X.
Examples
The order topology of a finite product of ordered vector spaces is identical to the product topology of the topological product of the constituent ordered vector spaces.
