Order Compactness in Riesz Spaces

Compact sets have important properties, and their studies have contributed to the development of functional analysis, particularly in the field of compact operators. In this paper, we introduce the concept of order compactness in Riesz spaces as an analog to topological compactness in the absence of a topology. We define order compact sets based on order convergence of nets and subnets, explore properties of these sets (e.g., closure, boundedness, preservation under order continuous maps), and prove results analogous to those in topological spaces, including an order version of the Banach-Stone theorem (Theorem 4.4) and a fixed point theorem (Theorem 4.4). When we introduce a continuous Banach lattice norm of order, we will show that the compactness of order coincides with the compactness of norm.