Abstract. By modifying the well-known concept of a fuzzy metric introduced in 1994 by A. George and P. Veeramani, we define the notion of a fragmentary fuzzy pseudometric. In this paper, we study basic properties of fragmentary fuzzy pseudometrics, describe the induced fuzzy supratopologies and fuzzy topologies, as well apply fragmentary fuzzy pseudometrics for the description of the analytic structure of the sets of right-infinite words.

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