Summary: The author introduces and studies the concepts “gradation of openness” and “fuzzy gradation of openness” on Boolean algebras. The notion of fuzzy topological Boolean algebra is defined and studied in [the author and K. N. Meenakshi, Indian J. Pure Appl. Math. 28, No. 12, 1639-1648 (1997; Zbl 0898.54007)].