In this paper, we introduce a notion of transfinite nuclear dimension for \(C^*\) -algebras, which coincides with the nuclear dimension when taking values in natural numbers. We use it to characterise a stronger form of having nuclear dimension at most \(\omega \) and moreover, we show that the transfinite nuclear dimension of a uniform Roe algebra is bounded by the transfinite asymptotic dimension of the underlying space. Hence we obtain that the uniform Roe algebra for spaces with asymptotic property C has the corona factorisation property.
