We present a complete, separable and metrizable topology on the product space of information and (subjective) beliefs. Such a topology formalizes similarity of differential information without the assumption of a common prior, but under the assumption that objectively impossible events are considered impossible by subjective beliefs. As an application to the theory of the consumer, we provide results on the continuity of expected utility and demand functions. We also provide continuity results for the value of information and the insurance premium as defined in the literature.