Order topology (functional analysis)

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In mathematics, specifically in order theory and functional analysis, the order topology of an ordered vector space ${\displaystyle (X,\leq )}$ is the finest locally convex topological vector space (TVS) topology on ${\displaystyle X}$ for which every order interval is bounded, where an order interval in ${\displaystyle X}$ is a set of the form ${\displaystyle [a,b]:=\left\{z\in X:a\leq z{\text{ and }}z\leq b\right\}}$ where ${\displaystyle a}$ and ${\displaystyle b}$ belong to ${\displaystyle X.}$^[1]

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 ${\displaystyle (X,\leq ),}$ rather than from some topology that ${\displaystyle X}$ starts out having. This allows for establishing intimate connections between this topology and the algebraic and order theoretic properties of ${\displaystyle (X,\leq ).}$ For many ordered topological vector spaces that occur in analysis, their topologies are identical to the order topology.^[2]