  For metric spaces, the doubling property, the uniform disconnectedness, and
the uniform perfectness are known as quasi-symmetric invariant properties. The
David-Semmes uniformization theorem states that if a compact metric space
satisfies all the three properties, then it is quasi-symmetrically equivalent
to the middle-third Cantor set. We say that a Cantor metric space is standard
if it satisfies all the three properties; otherwise, it is exotic. In this
paper, we conclude that for each of exotic types the class of all the conformal
gauges of Cantor metric spaces has continuum cardinality. As a byproduct of our
study, we state that there exists a Cantor metric space with prescribed
Hausdorff dimension and Assouad dimension.
