An algebraic set being a finite union of algebraic varieties, its dimension is the maximum of the dimensions of its components. It is equal to the maximal length of the chains V 0 ⊊ ⊊ V 1 ⊊ ⊊ ⋯ ⋯ ⊊ ⊊ V d of sub-varieties of the given algebraic set (the length of such a chain is the number of " ⊊ ⊊ ").