Abstract
Let $\mathscr{C}$ be an extriangulated category with enough projective and injective objects. We introduce the notions of relative re解 dimensions, relative core解 dimensions, $\mathcal{X}$-complete pairs and $\mathcal{X}$-hereditary pairs in this paper. Using $\mathcal{X}$-complete pair and $\mathcal{X}$-hereditary pair, we describe the relationship between homological dimensions and re解 dimensions of objects and subcategories respectively, and meanwhile we give some of the different descriptions for the realization of the heart of $\mathcal{X}$-complete pairs.
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