This paper reports some new results in relation to simplicial algorithms considering continuities of approximate fixed point sets. The upper semi-continuity of a set-valued mapping of approximate fixed points using vector-valued simplicial methods is proved, and thus one obtains the existence of finite essential connected components in approximate fixed point sets by vector-valued labels; examples are given to show that this is very different from the property for integer-valued labeling simplicial methods. The existence of essential sets is also proved focusing on both perturbations of domains and functions.