Human activity prediction has become a prerequisite for service recommendation and anomaly detection systems in a smart space including ambient assisted living (AAL) and activities of daily living (ADL). In this paper, we present a novel approach to predict the next-activity set in a multi-user smart space. Differing from the majority of the previous studies considering single-user activity patterns, our study considers multi-user activities that occur with a large variety of patterns. Its complexity increases exponentially according to the number of users. In the multi-user smart space, there can be inevitably multiple next-activity candidates after multi-user activities occur. To solve the next-activity problem in a multi-user situation, we propose activity set prediction rather than one activity prediction. We also propose activity sequence partitioning to reduce the complexity of the multi-user activity pattern. This divides an activity sequence into start, ongoing, and finish zones based on the features in the tendency of activity occurrences. The majority of the activities in a multi-user environment occur at the beginning or end, rather than the middle, of an activity sequence. Furthermore, the types of activities typically occurring in each zone can be sufficiently distinguishable. Exploiting these characteristics, we suggest a two-step procedure to predict the next-activity set utilizing a long short-term memory (LSTM) model. The first step identifies the zones to which current activities belong. In the next step, we construct three different LSTM models to predict the next-activity set in each zone. To evaluate the proposed approach, we experimented using a real dataset generated from our campus testbed. Our experiments confirmed the complexity reduction and high accuracy in the next-activity set prediction. Thus, it can be effectively utilized for various applications with context-awareness in a multi-user smart space.