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In numerous applications, for instance in predictive maintenance, there is a pression to predict events ahead of time with as much accuracy as possible while not delaying the decision unduly. This translates in the optimization of a trade-off between earliness and accuracy of the decisions, that has been the subject of research for time series of finite length and with a unique label. And this has led to powerful algorithms for Early Classification of Time Series (ECTS). This paper, for the first time, investigates such a trade-off when events of different classes occur in a streaming fashion, with no predefined end. In the Early Classification in Open Time Series problem (ECOTS), the task is to predict events, i.e. their class and time interval, at the moment that optimizes the accuracy vs. earliness trade-off. Interestingly, we find that ECTS algorithms can be sensibly adapted in a principled way to this new problem. We illustrate our methodology by transforming two state-of-the-art ECTS algorithms for the ECOTS scenario. Among the wide variety of applications that this new approach opens up, we develop a predictive maintenance use case that optimizes alarm triggering times, thus demonstrating the power of this new approach.