### Generalized fuzzy compactness in $L$-topological spaces.

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We prove by using well-founded trees that a countable product of supercomplete spaces, scattered with respect to Čech-complete subsets, is supercomplete. This result extends results given in [Alstera], [Friedlera], [Frolika], [HohtiPelantb], [Pelanta] and its proof improves that given in [HohtiPelantb].

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