The topology of the set of multiple zeta-star values

We provide a multiple integral representation for each multiple zeta-star value, and utilize these representations to establish a natural order structure on the set of such values. This order structure allows for a one-to-one correspondence between a subset of the infinite sequences of natural numbers and the half line $(1,+\mathrm{\infty })$. Some basic properties of this correspondence are discussed. We also calculate the Hausdorff dimensions for the images of some subsets of the infinite sequences under this correspondence. As a result of this correspondence, we are able to determine the limits for a number of natural multiple integrals. Our analysis also reveals that the set of multiple zeta-star values is dense within the $(1,+\mathrm{\infty })$ domain, and that each value is non-integer in nature.