Abstarct: In this thesis‎, ‎we first study some properties of best simultaneous approximation by thease sets‎. ‎Then We introduce a generalization concept of the best simultaneous approximation‎, ‎namely‎ ε-simultaneous approximation and prove some results about the properties‎ ‎of ε-simultaneous approximation with respect to downward sets in vector lattice Banach spaces‎. ‎