We present semi-algebraic transition systems endowed with metrics, which extend traditional transition systems by combining semi-algebraic sentences and metrics for measuring the deviation of system behaviors. In order to specify the distance between behaviors, we present two metrics over states and trajectories of semi-algebraic transition systems, respectively. Accordingly, we give an intuitive description of approximate satisfaction of system properties expressed as semi-algebraic sentences. Furthermore, we propose a concise definition of approximate semi-algebraic transition systems. Finally, we summarize our work with a conclusion implying that a criterion satisfying the transferability on transitions is required, which aims to construct approximate semi-algebraic transition systems with respect to general metrics.