Summary: In this paper, the closeness of the \(\tau\)-standard part of a set is discussed. Some related properties of the \(\tau\)-neighborhood system of a set are given. Then some related conclusions concerning the \(\tau\)-monad of a set and the \(\tau\)-standard part of a set are presented. Based on these results, necessary and sufficient conditions for the enlarged model and the saturated model are shown. Finally, some sufficient conditions on the \(\tau\)-standard part of a set to be closed are proved in the enlarged model and in the saturated model.