Consider a separable metric space (X, d), and let (K (X), d˜) denote the space of non-empty
compact subsets of X equipped with the Hausdorff metric. This paper aims to introduce and
investigate the concepts of two general fractal dimensions and general dimensions within
the framework of (K (X), d˜). In particular, we explore a relationship between the general
fractal dimensions of a set Z of a self-similar sequence space and their counterparts in the
space of compact subsets K (Z).