Jaccard similarity leads to the Marczewski-Steinhaus topology for information retrieval

摘要

We show that if the similarity function of a retrieval system leads to a (pseudo-) metric, the retrieval, the similarity and the Everett-Cater metric topology coincide and are generally different from the discrete topology. This is the case if we represent documents by lists and use the Jaccard similarity measure. The corresponding metric is then the Marczewski-Steinhaus metric. We further study the special case of a one-element query space consisting of a single-item query.