Determinization of finite automata is performed by the Subset Construction algorithm (SC). However, some application domains, including monitoring and diagnosis of active systems in artificial intelligence, and model-based mutation testing in software engineering, require determinization to be performed incrementally, in real time. Making incremental determinization by means of SC is bound to poor performances. To this end, an algorithm called Incremental Subset Construction (ISC) was proposed a few years ago. Disturbingly, this algorithm was recently discovered to be incorrect is some instance problems. The incorrect behavior of ISC originates when the redirection of a transition causes a portion of the DFA to be disconnected from the initial state. This paper provides a revisited version of ISC, called Revised Incremental Subset Construction (RISC), operating in either busy or lazy mode. In busy mode, the disconnection is always avoided. In lazy mode, since no check of disconnection is performed, a garbage collector is eventually required.