Order topology (functional analysis)

Mgkrupa: Created page

---

In [[order theory]] and [[functional analysis]], the '''order topology''' of an [[ordered vector space]] (''X'', ≤) is the finest [[locally convex]] [[topological vector space]] (TVS) [[topology]] on ''X'' for which every order interval is bounded, where an '''order interval''' in ''X'' is a set of the form [''a'', ''b''] := { ''z'' ∈ ''X'' : ''a'' ≤ ''z'' and ''z'' ≤ ''b'' } where ''a'' and ''b'' belong to ''X''.

The order topology is a vert important topology that is used frequently in the theory of [[ordered topological vector space]]s because the topology stems directly from the algebraic and order theoretic properties of (''X'', ≤), 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 (''X'', ≤).

== Definitions ==

The family of all locally convex topologies on ''X'' for which every order interval is bounded is non-empty (since it contains the coarsest possible topology on ''X'') 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 set|absorbing]] since [''x'', ''x''] = { ''x'' } for all ''x'' in ''X''.

== See also ==

* [[Ordered topological vector space]]
* [[Ordered vector space]]
* [[Vector lattice]]

== References ==


* <!-- -->



<!--- Categories --->

[[Category:Functional analysis]]

from Wikipedia - New pages [en] https://ift.tt/2XaeX68
via IFTTT