The purpose of this paper is to define and study some new types hesitant fuzzy open sets namely, α -open, preopen, semiopen, b-open β -open in topological space. Some properties the relationships between these are investigated. Furthermore, them subspace introduced.

By applying a recent result on the distributions of full and non-full words, we give proof useful folklore result: Hausdorff dimension an arbitrary set in shift space is equal to its natural projection [0, 1]. It has been used some former papers without proof. Then clarify that order calculate frequency sets using variational formulae, one only needs focus Markov measures explicit step when $$\...