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Many systems and strategies have been proposed for processing nonterminating data streams. Each approach has advantages and disadvantages, including the kinds of queries that can be executed. We present a framework for characterizing the kinds of queries that can be executed over streams based on a notion of compact sets from topology. We first apply our framework to queries over punctuated data streams. Previous work on punctuations focused primarily on the behavior of individual query operators. We use our framework to determine if an entire query can benefit from punctuations available from stream sources. We then consider other common strategies proposed in the literature for executing queries over streams, and we discuss how our framework can characterize the kinds of queries each strategy can answer.